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Characterization of the Distribution of Developmental Instability

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Characterization of the Distribution of Developmental Instability
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2008

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Asymmetry ( jstor )
Cotton ( jstor )
Drosophila ( jstor )
Kurtosis ( jstor )
Landmarks ( jstor )
Modeling ( jstor )
Population distributions ( jstor )
Population genetics ( jstor )
Population growth ( jstor )
Sample size ( jstor )

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CHARACTERIZATION OF THE DISTRIBUTION OF DEVELOPMENTAL
INSTABILITY













By

GREGORY ALAN BABBITT


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Gregory Alan Babbitt



























This document is dedicated to my grandmother, Sarah Miller, who taught me to admire
the natural world.















ACKNOWLEDGMENTS

I would like to thank Rebecca Kimball, Susan Halbert, Bernie Hauser, Jane

Brockmann, and Christian Klingenberg for helpful discussions and comments; Susan

Halbert and Gary Steck (Division of Plant Industry, State of Florida) for specimen ID;

Marta Wayne for use of her microscope and digital camera; Glenn Hall (Bee Lab,

University of Florida) for a plentiful supply of bees; and the Department of Entomology

and Nematology (University of Florida) for use of its environmental chambers. I thank

Callie Babbitt for help in manuscript preparation and much additional love and support.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES .................................................... ....... .. .............. viii

LIST OF FIGURES ......... ......................... ...... ........ ............ ix

ABSTRACT ........ ........................... .. ...... .......... .......... xii

CHAPTER

1 SEARCHING FOR A CONSISTENT INTERPRETATION OF
DEVELOPMENTAL INSTABILITY? A GENERAL INTRODUCTION ................1

2 ARE FLUCTUATING ASYMMETRY STUDIES ADEQUATELY SAMPLED?
IMPLICATIONS OF A NEW MODEL FOR SIZE DISTRIBUTION..................... 10

Intro du action .......................... ................... ... ... ................. ............ 10
Developmental Stability: Definition, Measurement, and Current Debate ..........10
The Distribution of Fluctuating Asymmetry .................................................12
M e th o d s ........................................................................... 1 7
R e su lts ...................................... .................................................... 2 2
D isc u ssio n ......................................................... .............. ................ 3 0
T he D distribution of F A ................................................................. .....................30
Sample Size and the Estimation of M ean FA................... .................................32
The Basis of Fluctuating Asymmetry..... .......... ...................................... 33
C o n c lu sio n .................................................. ................ 3 4

3 INBREEDING REDUCES POWER-LAW SCALING IN THE DISTRIBUTION
OF FLUCTUATING ASYMMETRY: AN EXPLANATION OF THE BASIS OF
DEVELOPMENTAL INSTABILITY .............................. .....................36

In tro d u ctio n ......................................................... ............. ................ 3 6
W hat is the B asis of FA ? .................................. ........... .................. 36
Exponential Growth and Non-Normal Distribution of FA..............................38
Testing a M odel for the Basis of FA ............................ .............. ............. 40
M e th o d s ..............................................................................4 1
M odel D evelopm ent ....... .......... ........ .................... .............. 41
Simulation of geometric Brownian motion................................ ............... 41









Simulation of fluctuating asymmetry .........................................................44
Inbreeding E xperim ent ............................................................. .....................4 5
M orphom etric analyses ............................................... ............................ 48
M odel selection and inference....................................... .......................... 49
R e su lts ...................................... .................................................... 4 9
M odel Sim ulation ...................... .................. ................... ..... .... 49
E x p erim ental R esu lts......................................... ............................................50
M odel Selection and Inference................................ ......................... ........ 52
D iscu ssion ................................................................................. 54
Revealing the Genetic Component of FA .............. .....................................54
Lim stations of the M odel ......................................................... .............. 56
T he Sources of Scaling .......................................................................... .... ... 56
Potential Application to Cancer Screening........................................................57
C o n c lu sio n .................................................. ................ 5 8

4 TEMPERATURE RESPONSE OF FLUCTUATING ASYMMETRY TO IN AN
APHID CLONE: A PROPOSAL FOR DETECTING SEXUAL SELECTION ON
DEVELOPM EN TAL IN STABILITY ........................................................................60

In tro d u ctio n ......................................................... ............. ................ 6 0
T he G enetic B asis of F A ......................................................................... ..... 60
The Environm ental Basis of FA ....................................................................... 61
Temperature and FA in and Aphid Clone ................................. ............... 63
M eth o d s ..............................................................................6 5
R e su lts ...................................... .................................................... 6 8
D iscu ssio n ...................................... ................................................. 7 2

5 CONCLUDING REMARKS AND RECOMMENDATIONS .............................. 75

H ow M any Sam ples Are Enough? ................................ .. ............................... 76
W hat M measure of FA is best?................ ......................................... ............... 76
Does rapid growth stabilize or destabilize development? .......................................78
Can fluctuating asymmetry be a sexually selected trait?................. .................78
Is fluctuating asymmetry a valuable environmental bioindicator?.............................79
Scaling Effects in Statistical Distributions: The Bigger Picture..............................80

APPENDIX

A LANDMARK WING VEIN INTERSECTIONS CHOSEN FOR ANALYSISOF
FLUCTUATING A SYM M ETRY ..................................... ......................... .......... 84

B USEFUL MATHEMATICAL FUNCTIONS ...................................... ...............86

A sym m etric Laplace D istribution......................................... .......................... 86
H alf-N orm al D distribution ........................................ ............................................86
L ognorm al D distribution ............................................................. ...........................86
Double Pareto Lognormal Distribution ............................. .................... 86









L IST O F R E F E R E N C E S ........................................................................ .. ....................88

B IO G R A PH IC A L SK E TCH ...................................................................... ..................96















LIST OF TABLES


Tablege

2-1. Maximized log-likelihood (MLL), number of model parameters (P) and Akaike
Information Criterion differences (A AIC) for all distributional models tested.
Winning models have AIC difference of zero. Models with nearly equivalent
goodness-of-fit to winners are underscored (A AIC <3.0). 4.0 indicates some support for specified model. A AIC >10.0 indicates no support
(Burnham and Anderson 1998). Distances between landmarks (LM) used for
first univariate size FA, aphids LM 1-2, bees LM 1-4 and long-legged fly LM 3-
6 and for second univariate FA aphids LM 2-3, bees LM 2-6 and long-legged fly
L M 4 -5 ............ .......... .... ................. .................................................2 3

2-2. Best fit parameters for models in Table 2-1. Parameters for univariate size FA
are very similar to multivariate size FA and are not shown. Skew indexes for
asymmetric Laplace are also not shown............... ................... ....... ........... 24

3-1. Distribution parameters and model fit for multivariate FA in two wild
populations and four inbred lines of Drosophila simulans and one isogenic line
of Drosophila melanogaster. Model fits are A AIC for unsigned centroid size
FA (zero is best fit, lowest number is next best fit). .............................................. 51















LIST OF FIGURES


Figure page

2-1. Schematic representation of mathematical relationships between candidate
models for the distribution of fluctuating asymmetry. Mixtures here are
continue ou s. ........................................................ ................. 14

2-2. Distribution of multivariate shape FA of A) cotton aphid (Aphis gossipyii) B)
domestic honeybee (Apis mellifera) and C) long-legged fly (Chrysosoma
crinitus). Best fitting lognormal (dashed line, lower inset), and double Pareto
lognormal (solid line, upper inset) are indicated................................................25

2-3. Distribution of multivariate centroid size FA of A) cotton aphid (Aphis gossipyii)
B) domestic honeybee (Apis mellifera) and C) long-legged fly (Chrysosoma
crinitus). Best fitting half-normal (dashed line, lower inset) and double Pareto
lognormal distribution (solid line, upper inset) are indicated. ................................26

2-4. Distribution of univariate unsigned size FA of A) cotton aphid (Aphis gossipyii)
B) domestic honeybee (Apis mellifera) and C) long-legged fly (Chrysosoma
crinitus). Best fitting half-normal (dashed line, lower inset) and double Pareto
lognormal distribution (solid line, upper inset) are indicated. ................................27

2-5. Distribution of sample sizes (n) from 229 fluctuating asymmetry studies reported
in three recent meta-analyses (Vollestad et al. 1999, Thornhill and Moller 1998
and Polak et al. 2003). Only five studies had sample sizes greater than 500 (not
show n). ...............................................................................28

2-6. Relationship between sample size and % error for estimates of mean FA drawn
from best fitting size (dashed line) and shape (solid line) distributions using
1000 draws per sample size. All runs use typical winning double Pareto
lognormal parameters (shape FA v = -3.7, T = 0.2, a = 1000, 0 = 9; for size FA v
= 1.2 = 0 .7 a = 4 .0 = 4 .0 ) .......................................................................... .....2 9

2-7. The proportion and percentage (inset) of individuals with visible developmental
errors on wings (phenodeviants) are shown for cotton aphids (A) and honeybees
(B) in relation to distribution of shape FA (Procrustes distance). Average FA for
both normal and phenodeviant aphids (C) and honeybees (D) are also given.........30

3-1. Ordinary Brownian motion (lower panel) in N simulated by summing
independent uniform random variables (W) (upper panel) ....................................42









3-2. Geometric Brownian motion in N and log N simulated by multiplying
independent uniform random variables. This was generated using Equation 1.5
w ith C = 0 .5 4 ...................................................................... 4 3

3-3. Geometric Brownian motion in N and log N with upward drift. This was
generated using Equation 1.5 with C = 0.60. ................................ ..................44

3-4. Model representation of Reed and Jorgensen's (2004) physical size distribution
model. Variable negative exponentially distributed stopping Times of random
proportional growth (GBM with C = 0.5) create double Pareto lognormal
distribution of size .............................. ................................... 46

3-5. A model representation of developmental instability. Normally variable
stopping times of random proportional growth (GBM with C > 0.5) create
double Pareto lognormal distribution of size. .................................. .................47

3-6. Simulated distributions of cell population size and FA for different amounts of
variation in the termination of growth (variance in normally distributed growth
stop time). The fit of simulated data to the normal distribution can determined
by how closely the plotted points follow the horizontal line (a good fit is
h horizontal ........................................................ ................ 50

3-7. Distribution of fluctuating asymmetry and detrended fit to normal for two
samples of wild population collected in Gainesville, FL in summers of 2004 and
2005 and four inbred lines of Drosophila simulans derived from eight
generations of full-sib crossing of the wild population of 2004. Also included is
one isogenic line of Drosophila melanogaster (me175). All n = 1000. The fit of
data to the normal distribution can determined by how closely the plotted points
follow the horizontal line (a good fit is horizontal). .............................................. 53

4-1. Predicted proximate and ultimate level correlations of temperature and growth
rate to fluctuating asymmetry are different. Ultimate level (evolutionary) effects
assume energetic limitation of individuals in the system. Proximate level
(growth mechanical) effects do not. Notice that temperature and fluctuating
asymmetry are negatively correlated in the proximate model while in the
ultimate model they are positively correlated. ................................. ............... 64

4-2. Cotton aphid mean development time +1 SE in days in relation to temperature
(n= 53 1). ...............................................................................68

4-3. Distribution of isogenic size, size based and shape based FA in monoclonal
cotton aphids grown in controlled environment at different temperatures.
Distributions within each temperature treatment are similar to overall
distributions show n here................................................. .............................. 69

4-4. Mean isogenic FA for (A.) centroid size- based and (B.) Procustes shape-based)
in monoclonal cotton aphids (collected in Gainesville FL) grown on isogenic
cotton seedlings at different temperatures.................... ........... .............. 70









4-5. Kurtosis of size-based FA in monoclonal cotton aphids grown on isogenic cotton
seedlings at different tem peratures ........................................................................ 71














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CHARACTERIZATION OF THE DISTRIBUTION OF DEVELOPMENTAL
INSTABILITY

By

Gregory Alan Babbitt

May 2006

Chair: Rebecca Kimball
Cochair: Benjamin Bolker
Major Department: Zoology

Previous work on fluctuating asymmetry (FA), a measure of developmental

instability, has highlighted its controversial relationship with environmental stress and

genetic architecture. I suggest that conflict may derive from the fact that the basis of FA

is poorly understood and, as a consequence, the methodology for FA studies may be

flawed. While size-based measures of FA have been assumed to have half-normal

distributions within populations, developmental modeling studies have suggested other

plausible distributions for FA. Support for a non-normal distribution of FA is further

supported by empirical studies that often record leptokurtic (i.e., fat or long-tailed)

distributions of FA as well. In this dissertation, I investigate a series of questions

regarding the both the basis and distribution of FA in populations. Is FA normally

distributed and therefore likely to be properly sampled in FA studies? If not normal,

what candidate model distribution best fits the distribution of FA? Is the shape of the

distribution of FA similar to a simple and specific growth model (geometric Brownian









motion)? Does reducing individual variation in populations through inbreeding affect

follow the prediction of this model? How does this shape respond to environmental

factors such as temperature when genetic variation is controlled?

In three species of insects (cotton aphid, Aphis gossipyii Glover; honeybee, Apis

mellifera; and long-legged fly, Chrysosoma crinitus (Dolichipodidae)), I find that FA was

best described by a double Pareto lognormal distribution (DPLN), a lognormal

distribution with power-law tails. The large variance in FA under this distribution and

the scaling in the tails both act to slow convergence to the mean, suggesting that many

past FA studies are under-sampled when the distribution of FA is assumed to be normal.

Because DPLN can be generated by geometric Brownian motion, it is ideal for describing

behavior of cell populations in growing tissue. I demonstrated through both a

mathematical growth model and an inbreeding experiment in Drosophila simulans that

the shape of the distribution of FA is highly dependent on the level of genetic redundancy

or heterogeneity in a population. In monoclonal lines of cotton aphids, I also demonstrate

that FA decreases with temperature and that a shift in kurtosis is associated with

temperature induced phenotypic plasticity. This supports the prediction of a proximate

model for the basis of FA and also suggests shape of the distribution of FA responds to

environmentally induced changes in gene expression on the same genetic background.














CHAPTER 1
SEARCHING FOR A CONSISTENT INTERPRETATION OF DEVELOPMENTAL
INSTABILITY? A GENERAL INTRODUCTION

Everywhere, nature works true to scale and everything has its proper size.
-- D'Arcy Thompson

There have been many incarnations of the idea that stability and symmetry are

somehow related. Hippocrates (460-377 B.C.) was the first to postulate internal

corrective properties that work in the presence of disease. Waddington (1942) suggested

existence of similar homeostatic buffering against random and presumably additive errors

occurring during development. The term "fluctuating asymmetry," first introduced by

Ludwig (1932), was later adopted by Mather (1953), Reeve (1960) and Van Valen (1962)

to describe a measurable form of morphological noise representing a hypothetical lack of

buffering that is always present during development of organisms. Recently, biologists

have become very interested both in fluctuating asymmetry's potential usefulness as a

universal bioindicator of environmental health (Parsons 1992) and in its potential as an

indicator or even an overt signal of an individual's overall genetic quality (Moller 1990).

However, over a decade of work has left the field with no clear relationship between

increased fluctuating asymmetry and either environmental or genetic stress (Bjorksten et

al. 2000, Lens et al. 2002). Despite this fact, fluctuating asymmetry is still often assumed

to indicate of developmental instability. Recently, Debat and David (2001) page 560

define developmental stability as "a set of mechanisms historically selected to keep the

phenotype constant in spite of small random developmental irregularities potentially









inducing slight differences among homologous parts within individuals." (I have

italicized the aspects of this definition that I think are left undefined.)

Fluctuating asymmetry is defined and measured as the average right minus left

difference in size or shape of morphological characters in a population and has been

generally accepted as an indicator of developmental instability because both sides of a

bilaterally symmetric organism have been developed by the same genetic program in the

same environment (Moller and Swaddle 1997). Fluctuating asymmetry is measured as

left-right side differences in the size or shape of paired bilaterally symmetric biological

structures of organisms. In a character trait that demonstrates fluctuating asymmetry, it is

assumed that the distribution of signed left-right differences is near zero and that there is

no selection for asymmetry (Palmer and Strobek 1986, 2003). Other types of asymmetry

do exist and are thought to indicate selection against symmetry. Directional asymmetry

denotes a bias towards left or right sidedness that causes the population mean to move

away from zero. In antisymmetry, left or right side biases occur equally at the individual

level creating a population that is bimodal or platykurtic.

While much attention has been directed toward the possible genetic basis of

fluctuating asymmetry (reviewed by Leamy and Klingenberg 2005, Woolf and Markow

2003), response of fluctuating asymmetry to stress (reviewed by Hoffman and Woods

2003) and correlation of fluctuating asymmetry with mate choice (Moller and Swaddle

1997); little scientific effort has been directed toward investigating its basis or origin at

levels of organization lower than the individual. A few theoretical explanations for the

basis of fluctuating asymmetry have been developed (reviewed by Klingenberg 2003).

Only two of these offer a causal explanation for the increased levels of fluctuating









asymmetry that are sometimes observed under periods of environmental stress. While

not mutually exclusive, these two explanations differ at the organizational level at which

they are thought to act. Moller and Pomiankowski (1993) first suggested that strong

natural or sexual selection can remove regulatory steps controlling the symmetric

development of certain traits (e.g., morphology used in sexual display). They suggest that

with respect to sexually selected traits (and assuming that they are somehow costly to

produce), individuals may vary in their ability to buffer against environmental stress in

proportion to the size of their own energetic reserves. These reserves are often indicative

of the genetic quality of individuals. Therefore, high genetic quality is expected to be

associated with low fluctuating asymmetry. Emlen et al. (1993) present another and

more proximate explanation of the basis of fluctuating asymmetry. They do not invoke

sexual selection but hypothesize that fluctuating asymmetry is due largely to the non-

linear dynamics of signaling and supply that may occur during growth. Here fluctuating

asymmetry is thought to result from the scaling up of compounding temporal

asymmetries in signaling between cells during growth. In their model, hypothetical

levels of signaling compounds (morphogens) and or growth precursors used in the

construction of cells vary randomly over time. When growth suffers less interruption, in

other words, when it occurs faster and under less stress, there is also less complexity (and

fractal dimension) in the dynamics of signaling and supply. This is should reduce

fluctuating asymmetry.

Only a few other models have since been proposed. A model by Graham et al.

(1993) suggests that fluctuating asymmetry in the individual can also be the net result of

compounding time lags and chaotic behavior between hormonally controlled growth rates









on both sides of an axis of bilateral symmetry. More recently Klingenberg and Nijhout

(1999) present a model of morphogen diffusion and threshold response that includes

genetic control of each component. They demonstrate that fluctuating asymmetry can

result from genetically modulated expression of variation that is entirely non-genetic in

origin. In other words, even without specific genes for fluctuating asymmetry,

interaction between genetic and non-genetic sources of variation (G x E) can cause

fluctuating asymmetry (Klingenberg 2003).

While all these theories for the basis of fluctuating asymmetry have proven

useful in making some predictions about fluctuating asymmetry in relation to sexual

selection and growth rate/trait size, none are grounded in any known molecular

mechanisms. The search for any single molecular mechanism that stabilizes the

developmental process has proven elusive. A recent candidate was the heat shock

protein, Hsp90, which normally target conformationally plastic proteins that act as signal

tranducers (i.e., molecular switches) in many developmental pathways (Rutherford and

Lundquist 1998). Because Hsp90 recognizes protein folding, it can also be diverted to

misfolded proteins that are denatured during environmental stress. Therefore, Hsp90 can

potentially link the developmental process to the environment and these authors suggest it

may also capacitate the evolution of novel morphology during times of stress by

revealing genetic variation previously hidden to selection in non-stressful environments.

However, additional research by Milton et al. (2003) shows that while Hsp90 does buffer

against a wide range of morphologic changes and does mask the effect of much hidden

genetic variation in Drosophila, it does not appear to affect average levels of fluctuating

asymmetry through any single Hsp90 dependent pathway or process.









It is said that wisdom begins with the naming of things. The case of fluctuating

asymmetry reminds us that the act of giving names to things in science can, in fact, lend a

false impression that we have achieved a true understanding of that which has been

named. While fluctuating asymmetry has had several specific descriptive definitions, we

do not really know how to define it at a fundamental level because we do not understand

exactly how development is destabilized under certain conditions of both gene and

environment. All we can say for now is that fluctuating asymmetry is a mysterious form

of morphological variation. Mary Jane West-Eberhard (2003) has dubbed it the "dark

side" of variation because it may represent that noisy fraction of the physical-biological

interface that is still free of selection, and not under direct control of the gene.

Any study of natural variation would do well to begin by simply observing its

shape or its distribution in full. In nature, statistical distributions come mostly in two

flavors: those generated by large systems of independent additive components and those

generated by large systems of interacting multiplicative components (Vicsek 2001,

Sornette 2003). When large systems are composed of independent subunits, random

processes result in the normal distribution, the cornerstone of classical statistics. The

normal distribution is both unique and extreme in its rapidly decaying tails, its very

strong central tendency and its sufficient description by just two parameters, the mean

and variance, making it, in this sense, the most parsimonious description of random

variation. However, the normal distribution does not describe all kinds of stochastic or

random behavior commonly observed in the natural sciences. When subunits comprising

large systems interact, random processes are best described by models that underlie

statistical physics (sometimes called Levy statistics) (Bardou et al. 2003. Sornette 2003).









Fat-tailed distributions are often the result of propagation of error in the presence of

strong interaction. These types of distributions include the power function, Pareto, Zipf

and the double Pareto or log-Laplace distributions, all characterized by a power law and

an independence of scale. Collective effects in complex interacting systems are also

often characterized by these power laws (Wilson 1979, Stanley 1995). Examples include

higher order phase transitions, self-organized criticality and percolation. During second

order phase transition at the critical temperature between physical phases, external

perturbation of the network of microscopic interactions between molecules results in

system reorganization at a macroscopic level far above that of interacting molecules (e.g.,

the change from water to ice). This results in collective imitation that propagates among

neighboring molecules over long distances. Exactly at these critical temperatures,

imitation between neighbors can be observed at all scales creating regions of similar

behavior that occur at all sizes (Wilson 1979). Thus, a self-similar power-law manifests

itself in the interacting system's susceptibility to perturbation and results, in this case,

from the multiplicity of interaction paths in the interaction network (Stanley 1995). As

the distance between two objects in a network increases, the number of potential

interaction pathways increases exponentially and the correlation between such paths

decreases exponentially. The constant continuous degree of change represented by the

power law is the result of a combined effect between both an exponentially increasing

and decreasing rate of change. This highlights the fact that power laws can be easily

manifested from combinations of exponential functions which are very common to

patterns of change in many natural populations (both living and non-living).









Therefore, it is important to note that power laws in statistical distributions do not

have to always indicate strong interaction in a system and that many other simple

mechanisms can create them (Sornette 2003), especially where the behavior of natural

populations are concerned. For example, an apparently common method by which power

laws are generated in nature is when stochastic proportional (geometric) growth is

observed randomly in time (Reed 2001). Power law size distributions in particle size,

human population size, and economic factors are all potentially explained by this process

(Reed and Jorgensen 2004). Here we also have exponential increase in size opposing an

exponential decrease in the probability of observation or termination that results in a

gently decaying power law. This process can also explain why power law scaling can

occur through the mixing of certain distributions where locally exponentially increasing

and decreasing distributions overlap. For example, superimposing lognormal

distributions results in a lognormal distribution with power law tails (Montroll and

Shlesinger 1982,1983).

Given that exponential relationships are so common in the natural world, we should

assume that in observing any large natural population outside of an experiment, there is

probably some potential for a power-law scaling effect to occur. Therefore some degree

of non-normal behavior may be likely to be observed, often in the underlying

distribution's tail. If we assume an underlying normal distribution, and sample it

accordingly, we are likely to under-sample this tail. And so we rarely ever present

ourselves with enough data to challenge our assumption of normality, and we risk

missing the chance to observe a potentially important aspect of natural variation.









Until now, the basis of fluctuating asymmetry has been addressed only with very

abstract models of hypothetical cell signaling, or at the level of selection working on the

organism with potential mechanism remaining hypothetical. In this dissertation, the

underlying common theme is that fluctuating asymmetry must first and foremost be

envisioned as a stochastic process occurring during tissue growth, or in other words,

occurring within an exponentially expanding population of cells. This expansion process

can be modeled by stochastic proportional (geometric) growth that is terminated or

observed randomly over time. As will be explained in subsequent chapters, this

generative process can naturally lead to variation that distributes according to a

lognormal distribution with power laws in both tails (Reed 2001). In chapter 2, I

examine the distribution of fluctuating asymmetry in the wings of three species of insects

(cotton aphid, Aphis gossipyi Glover, honeybee, Apis mellifera, and long-legged flies,

Chrysosoma crinitus) and test various candidate models that might describe the statistical

distribution of fluctuating asymmetry. I then address whether, given the best candidate

model, fluctuating asymmetry studies have been appropriately sampled. I suggest that

much of the current controversy over fluctuating asymmetry may be due to the fact that

past studies have been under-sampled. In chapter 3, I extend and test a

phenomenological model for fluctuating asymmetry that is introduced in chapter 2. I

present the model and then examine some of its unique predictions concerning the effects

of inbreeding on the shape of the distribution of fluctuating asymmetry in Drosophila. I

present evidence that the genetic structure of a population can have a profound effect on

the scaling and shape of the observed distribution of fluctuating asymmetry. In chapter 4,

I characterize the pattern of developmental noise (fluctuating asymmetry in the absence






9


of genetic variation) in two monoclonal populations of cotton aphid cultured under

graded environmental temperatures. I investigate how developmental noise is altered by

this simple change in the environment. I present evidence that the environmental

response of size-based FA is directly related to developmental time. Lastly, I conclude

with a review of my major findings in the context of the introduction I have presented

here.














CHAPTER 2
ARE FLUCTUATING ASYMMETRY STUDIES ADEQUATELY SAMPLED?
IMPLICATIONS OF A NEW MODEL FOR SIZE DISTRIBUTION

Introduction

Developmental Stability: Definition, Measurement, and Current Debate

Developmental stability is maintained by an unknown set of mechanisms that

buffer the phenotype against small random perturbations during development (Debat and

David 2001). Fluctuating asymmetry (FA), the most commonly used assay of

developmental instability, is defined either as the average deviation of multiple traits

within a single individual (Van Valen 1962) or the deviation of a single trait within a

population (Palmer and Strobek 1986, 2003; Parsons 1992) from perfect bilateral

symmetry. Ultimately, an individual's developmental stability is the collective result of

random noise, environmental influences, and the exact genetic architecture underlying the

developmental processes in that individual (Klingenberg 2003; Palmer and Strobek

1986). Extending this to a population, developmental stability is the result of individual

variation within each of these three components.

Currently, there is conflict in the literature regarding the effect of both environment

and genes on the developmental stability of populations. The development of bilateral

symmetry appears to be destabilized to various degrees by both environmental stressors

(review in Moller and Swaddle 1997) and certain genetic architectures (usually created

by inbreeding: Graham 1992; Lerner 1954; Mather 1953; Messier and Mitton 1996;

review by Mitton and Grant 1984). While the influence of inbreeding on FA is not









consistent (Radwan 2003; Carchini et al. 2001; Fowler and Whitlock 1994; Leary et al.

1983,1984; Lens et al. 2002; Perfectti and Camachi 1999; Rao et al. 2002; Vollestad et al.

1999), it has led biologists to use terminology such as "genetic stress" or "developmental

stress" when describing inbred populations (Clarke et al. 1986 and 1993; Koehn and

Bayne 1989; Palmer and Strobek 1986).

While genetic and environmental stressors have been shown to contribute to

developmental instability and FA, the full picture is still unclear (Bjorksten et al. 2000;

Lens et al. 2002). While FA has been proposed as a universal indicator of stress within

individual organisms (Parsons 1992), its utility as a general indicator of environmental

stress has been contentious (Bjorksten et al. 2000; Ditchkoff et al. 2001; McCoy and

Harris 2003; Merila and Bjorklund 1995; Moller 1990; Rasmuson 2002; Thornhill and

Moller 1998; Watson and Thornhill 1994; Whitlock 1998). Despite many studies, no

clear general relationship between environmental stress and FA has been demonstrated or

replicated through experimentation across different taxa (Bjorksten et al. 2000; Lens et al.

2002). Furthermore, the effects of stress on FA appear to be not only species-specific but

also trait-specific and stress-specific (Bjorksten et al. 2000). Several meta-analyses have

attempted to unify individual studies on the relation of sexual selection, heterozygosity,

and trait specificity to FA (Polak et al. 2003; Thornhill and Moller 1998; Vollestad et al.

1999); while some weak general effects have been found, their biological significance is

still unresolved.

Taken together, the ambiguity of the results from FA studies suggests unresolved

problems regarding the definition and/or measurement of FA. The distribution and

overall variability of FA are sometimes discussed with regards to repeatability in FA









studies (Whitlock 1998; Palmer and Strobeck 2003), but is seldom a primary target of

investigation. Until we can quantify FA more reliably and understand its statistical

properties, the potential for misinterpretation of FA is likely to persist.

The Distribution of Fluctuating Asymmetry

Although it is always risky to infer underlying processes from observed patterns,

careful examination of the distribution of FA in large samples may help distinguish

between possible scenarios driving FA. For instance, a good fit to a single statistical

distribution may imply that the same process operates to create FA in all individuals in a

population. In contrast, a good fit to a discrete mixture of several different density

functions might suggest that highly asymmetric individuals suffer from fundamentally

different developmental processes than their more symmetric counterparts. Thin-tailed

distributions (e.g., normal or exponential) may indicate relative independence in the

accumulation of small random developmental errors, whereas heavy-tailed distributions

may implicate non-independent cascades in the propagation of such error. Despite much

interest in the relationship between environmental stress and levels of FA, the basic

patterns of its distribution in populations remain largely unexplored.

One common distributional attribute of FA, leptokurtosis, has been discussed in the

literature. Leptokurtosis denotes a distribution that has many small and many extreme

values, relative to the normal distribution. Two primary causes of this kind of departure

from the normal distribution are the mixing of distributions and/or scaling effects in data.

For example, the Laplace or double exponential distribution is leptokurtic (but not heavy-

tailed) and can be represented as a continuous mixture of normal distributions (Kotz et al.

2001; Kozubowski and Podgorski 2001). Just as log scaling in the normal distribution

results in the lognormal distribution, log scaling in the Laplace leads to log-Laplace (also









called double Pareto) distributions (Kozubowski and Podgorski 2002; Reed 2001), which

are both leptokurtic and heavy-tailed (see Figure 2-1). Several explanations for

leptokurtosis in the distribution of FA have been proposed. Both individual differences

in developmental stability within a population (Gangestad and Thornhill 1999) and

differences in FA between subpopulations (Houle 2000) have been suggested to lead to

continuous or discrete mixtures of normal distributions with different developmental

variances, which in turn would cause leptokurtosis (e.g., a Laplace distribution) in the

overall distribution of FA. Mixtures of non-normal distributions may also cause either

leptokurtosis or platykurtosis (more intermediate values than the normal distribution:

Palmer and Strobek 2003). A potential example of this is illustrated by Hardersen and

Frampton's (2003) demonstration that a positive relationship between mortality and

asymmetry can cause leptokurtosis. Alternatively, Graham et al. (2003) have argued that

developmental error should behave multiplicatively in actively growing tissues, creating

a lognormal size distribution in most traits rather than the normal distribution that is

usually assumed. They argue that this ultimately results in leptokurtosis (but not fat tails)

and size dependent expression of FA. Because simple growth models are often geometric,

we should not be surprised if distributions of size-based FA followed the lognormal

distribution (see Limpert et al. 2001 for a review of lognormal distributions in sciences).

Not well recognized within biology is the fact that close interaction of many

components can result in power-law scaling distributionall tails that decrease

proportional to x-a rather than to some exponential function of x such as exp(-ax)

[exponential] or exp(-ax2) [normal] and hence to heavy-tailed distributions (Sornette

2003). Power- law scaling is often associated with the tail of the lognormal distribution,






14


especially when log standard deviation is large ( Mitzenmacher unpublished ms.;

Montroll and Shlesinger 1982, 1983; Roman and Porto 2001; Romeo et al. 2003).


additive/linear


geometric/logarithmic


NORMAL


Normal-
Laplace


gA


Double Pareto
lognormal (DPLN)

:t


LAPLACE


LOGNORMAL


M
x
X
T
U
R
E


LOG-LAPLACE


Figure 2-1. Schematic representation of mathematical relationships between candidate
models for the distribution of fluctuating asymmetry. Mixtures here are
continuous.

KEY:
Solid line scale of variable (x *- ln(x))
Dashed line random walk observed at constant stopping rate (i.e., negative
exponentially distributed stop times) Note: random walk on log scale exhibits geometric
Brownian motion
Dotted line convolution of two distributions (one of each type)
Block arrow a continuous mixture of distributions with stochastic (exponentially
distributed) variance
(Note Log-Laplace is also called double Pareto by Reed 2001, Reed and Jorgensen
2004)


* *









Because FA in a population may reflect fundamental developmental differences

between different classes or groups of individuals, for example stressed and non-stressed,

or between different subpopulations as suggested by Houle (2000), we might expect

discrete mixtures of different distributions to best describe FA. For instance, extreme

individuals falling in a heavy upper tail may be those who have exceeded some

developmental threshold. Major disruption of development, resulting in high FA, may

also reveal the scaling that exists in the underlying gene regulatory network (Albert and

Barabasi 2002; Clipsham et al. 2002; Olvai and Barabasi 2002). Alternatively, ifFA is

produced by a single process, but to various degrees in different individuals, then one

might expect a continuous mixture model to best describe the distribution of FA.

The possibility of non-normal distribution of FA opens the door to several potential

sampling problems. For instance, if the lognormal shape, or multiplicative variance

parameter, is large, then broad distribution effects may slow the convergence of the

sample mean to the population mean as sample sizes are increased (Romeo et al. 2003).

An additional, thornier, problem is caused by power-law scaling in the tails of

distributions. Many lognormally distributed datasets exhibit power-law scaling (or

amplification) in the tail region (Montroll and Shlesinger 1982, 1983; Romeo et al. 2003;

Sornette 2003), sometimes called Pareto-Levy tails or just Levy tails. As sample sizes

grow infinitely large, power-law and Pareto distributions may approach infinite mean (if

the scaling exponent is less than three) and infinite variance (if the scaling exponent is

less than two), and therefore will not obey the Law of Large Numbers (that sample means

approach the population mean as sample sizes increase). Increased sampling actually

increases the likelihood of sampling a larger value in the tail of a Pareto distribution









(Bardou et al. 2003; Quandt 1966), creating more uncertainty in estimates of the mean as

sample size increases. The presence of power-law tails can slow overall convergence

considerably even in distributions that are otherwise lognormal with low variance (which

may not look very different from well-behaved lognormal distributions unless a large

amount of data is accumulated).

This discussion points to two effects that need to be assessed broad distribution

effects, controlled by the shape (lognormal variance) parameter of the body of the FA

distribution, and power-law tails, controlled by the scaling exponents of the tails of the

FA distribution. To assess these effects, I apply a new statistical model, the double

Pareto lognormal (DPLN) distribution (Reed and Jorgensen 2004). The DPLN

distribution is a lognormal distribution with power-law behavior in both tails (for values

near zero and large positive values). Similar to the log-Laplace distributions, the DPLN

distribution can be represented as a continuous mixture of lognormal distributions with

different variances. It can also be derived from a geometric Brownian motion (a

multiplicative random walk) that is stopped or "killed" at a constant rate (i.e., the

distribution of stop times is exponentially distributed: Reed and Jorgensen 2004; Sornette

2003). The parameters of the DPLN distribution include a lognormal mean (v) and

variance ('2) parameter which control the location and spread of the body of the

distribution, and power-law scaling exponents for the left (3) and right (c) tails. Special

cases of the DPLN include the right Pareto lognormal (RPLN) distribution, with a power-

law tail on the right but not the left side (->oo); the left Pareto lognormal (LPLN)

distribution, with a power-law tail only near zero (ca-c); and the lognormal distribution,

with no power-law tails (a->c,3->co). For comparison, I also fit normal and half-normal









distributions as well as the asymmetric Laplace distribution to the data on FA. See

Figure 2-1 for schematic representation of relationships between these candidate models

for FA.

In the following study, I directly test the fit of different distributions to large FA

datasets from three species of insects. I include a lab cultured monoclonal line of cotton

aphid, Aphis gossipyii Glover, in an attempt to isolate the distribution of developmental

noise for the first time. I also analyze data from a semi-wild population of domestic

honeybee, Apis mellifera, taken from a single inseminated single queen colony, and from

a large sample of unrelated wild-trapped Long-legged flies (Chrysosoma crinitus:

Dolichopididae).

I address two primary groups of goals in this study. First, I investigate what

distributions fit FA data best and how the parameters of these distributions vary across

species, rearing conditions, and levels of genetic relatedness. I also address whether

"outliers" (individuals with visible developmental errors) appear to result from discrete or

continuous processes. Secondly, I determine how accurate the estimates of population

mean FA are at various sample sizes to determine whether past studies of FA been

adequately sampled to accurately estimate mean or average FA in populations. In

addition to these two primary goals, I also compare the best-fitting distributions and level

of sampling error for three of the most common methods of measuring FA: a univariate

and a multivariate size-based metric of asymmetry, and a multivariate shape-based

method.

Methods

Wings were collected and removed from three populations of insects and dry

mounted on microscope slides. These populations included a monoclonal population of









1022 Cotton Aphid (Aphis gossipyii Glover) started from a single individual collected

from citrus in Lake Alfred, Florida, 1001 honeybees Apis mellifera, maintained at

University of Florida and 889 long-legged flies (Chrysosoma crinitus :Dolichopodidae).

All species identifications were made through the State of Florida Department of Plant

Industry in Gainesville and voucher specimens remain available in their collections.

Aphid cultures were maintained on potted plants in reach-in environmental

chambers at 150C with constant 14/10 hour LD cycle generated by 4 20 watt

fluorescent Grolux brand bulbs. Aphid cultures were cultured on approximately 10 day

old cotton seedlings (Gossipium) and allowed to propagate until crowded. Crowding

stimulated alate formation (winged forms) in later generations which were collected

every twenty days with a fine camel hair brush wetted in ethanol. New plants were added

every ten days and alates were allowed to move freely from plant to plant starting new

clones until they were collected. The temperature at which the colony was maintained

created a low temperature "dark morph" Cotton Aphid which still propagated on host

plants quickly but was larger than high temperature "light morphs" that form at

temperatures greater than 17C. Dark morphs colonize stems on cotton whereas light

morphs colonized the undersides of leaves.

The bees were collected in June 2004 by Dr. Glenn Hall at the University of

Florida's Bee Lab from a single inseminated single queen colony. They were presumably

all foragers and haplodiploid sisters collected as they exited the hive into a collection bag.

The bag was frozen for three hours and then the bees were placed in 85% ethanol.

Long-legged flies were trapped from a wild population using 14 yellow plastic

water pan traps in southwest Gainesville, Florida, during May 2003 and May-June 2004.









A very small amount of dishwashing detergent was added to the water to eliminate

surface tension and enhance trapping. Traps were checked every three hours during

daylight and set up fresh every day of trapping.

All insect specimens were dried in 85% ethanol, and then pairs of wings were

dissected (in ethanol) and air-dried to the glass slides while the ethanol evaporated.

Permount was used to attach cover slips. This technique prevented wings from floating

up during mounting, which might slightly distort the landmark configuration. Dry

mounts were digitally photographed. All landmarks were identified as wing vein

intersections on the digital images (six landmarks on each wing for aphids, eight for

honeybees and Dolichopodid flies). See Appendix A for landmark locations on wings for

each species.

Wing vein intersections were digitized three times each on all specimens using

TPSDIG version 1.31 (Rohlf, 1999). All measures of FA were taken as the average FA

value of the three replicate measurements for each specimen. Specimens damaged at or

near any landmarks were discarded. Fluctuating asymmetry was measured in three ways

on all specimens. First, a common univariate metric of absolute unsigned asymmetry

was taken for two different landmarks: FA = abs(R L) where R and L are the Euclidean

distances between the same two landmarks on either wing. In aphids, landmarks 1-2 and

2-3 were used; in bees, landmarks 1-4 and 2-6 were used; and in long-legged flies,

landmarks 3-6 and 4-5 were used. Two multivariate geometric morphometrics using

landmark-based methods were performed using all landmarks shown in Appendix A. A

multivariate size-based FA (FA 1 in Palmer and Strobek 2003) was calculated as absolute

value of (R L) where R and L are the centroid sizes of each wing (i.e., the average of









the distances of each landmark to their combined center of mass or centroid location). In

addition, a multivariate shape-based measure of FA known as the Procrustes distance was

calculated as the square root of the sum of all squared Euclidean distances between each

left and right landmark after two-dimensional Procrustes fitting of the data (Bookstein

1991; Klingenberg and McIntyre 1998; FA 18 in Palmer and Strobeck 2003; Smith et al.

1997). Procrustes fitting is a three step process including a normalization for centroid

size followed by superimposition of two sets of landmarks (right and left) and rotation

until all distances between each landmark set is minimized. Centroid size calculation,

Euclidean distance calculation and Procrustes fitting were performed using 0yvind

Hammer's Paleontological Statistics program PAST version 0.98 (Hammer 2002). For

assessing measurement error (ME) of FA (or more specifically, the digitizing error), we

conducted a Procrustes ANOVA (in Microsoft Excel) on all pairs of wing images

resampled three times each for every species (Klingenberg and McIntyre 1998). Percent

measurement error was computed as (ME/average FA) x 100 where

ME = (FA- FA2 + FA2 FA3 + FA FA3)/ 3 All subsequent statistical analyses

were performed using SPSS Base 8.0 statistical software (SPSS Inc.).

The fits of all measures of FA to eight distributional models (normal, half-normal,

lognormal, asymmetric Laplace, double Pareto lognormal (DPLN), two limiting forms of

DPLN, the right Pareto lognormal (RPLN) and the left Pareto lognormal (LPLN) and a

discrete mixture of lognormal and Pareto) were compared by calculating negative log

likelihood and Likelihood Ratio Test (LRT) if models were nested and Akaike

Information Criteria (AIC) if not nested (Burnham and Anderson 1998; Hilborn and

Mangel 1998). Both of these approaches penalize more complex models (those with









more parameters) when selecting the best-fit distributional model for a given dataset.

(The Likelihood Ratio Test does not technically apply when the nesting parameter is at

the boundary of its allowed region, e.g., when c->oc for the DPLN, but Pinheiro and

Bates (2000) suggest that the LRT is conservative, favoring simpler models, under these

conditions.) Best fitting parameters were obtained by maximizing the log-likelihood

function for each model (Appendix B). The maximization was performed using the

conjugate gradient method within unconstrained solve blocks in the program MathCad by

MathSoft Engineering and Education Inc (2001), and was also confirmed using Nelder-

Mead simplex algorithm or quasi-Newton methods in R version 2.0.1 (2003), a

programming environment for data analysis and graphics.

Phenodeviants were defined as individuals demonstrating missing wing veins, extra

wing veins or partial wing veins on either one or both wings. All phenodeviants in

honeybees involved absence of the vein at landmark 6 (LM 6). Phenodeviants in aphids

were more variable but mostly involved absence of wing vein intersections at LM 2 or

LM 3. Procrustes distances were estimated for phenodeviants by omitting the missing

landmarks (caused by the phenodeviance) and controlling for the effect of this removal

on the sums of squares. I added an average of the remaining sums of squares in place of

the missing sums of squares so that the calculated Procrustes distance is comparable to

normal specimens (i.e., six landmarks). In almost all phenodeviants, this involved

omission of only one set of landmark values. The frequency of phenodeviants was

examined across the range of the FA distribution (i.e., Procrustes distance), and mean

values of the FA for phenodeviants were compared to normal individuals in order to









assess whether phenodeviants tended to show higher than normal levels of FA in

characters that were not affected by the missing, partial, or extra wing veins.

The best fitting parameters of the best fitting models were used to build a

distributional model under which repeated sampling was simulated at various sample

sizes. Average error in estimation of the mean FA was calculated as a coefficient of

variation ((s / x) 100) for 1,000 randomly generated datasets. Lastly, comparison were

made of the estimation errors given the best fitting distributions of FA to a distribution of

sample sizes from 229 FA studies published in three recent meta-analyses (Polak et al.

2003; Thornhill and Moller 1998; Vollestad et al. 1999).

Results

In the distributions of shape-based FA in monoclonal cotton aphids (n = 1022),

domesticated honeybees (n = 1001), and wild trapped long-legged flies (n = 889), AIC

and LRTs always favored DPLN or RPLN models by a large margin. All size-based FA

distributions favored DPLN or LPLN by a large margin (Table 2-1). All variants of

discrete mixture models we tried had very poor results (data not shown). Figures 2-2

through 2-4 demonstrate best fitting models for multivariate shape FA (DPLN and

lognormal), multivariate centroid size FA (DPLN and half-normal), and univariate size

FA (DPLN and half-normal) for all three species. FA was often visually noticeable in

aphids, where the mean shape FA (Procrustes distance) was three times higher (0.062 +

0.00050) than in bees (0.023 0.00026) or flies (0.019 0.00028). I note that

distribution of size FA in aphids and bees fit half-normal distribution in the upper tails

fairly well but fit relatively poorly among individuals with low FA. Long-legged flies

exhibit poor fit to half-normal in both tails.












Table 2-1. Maximized log-likelihood (MLL), number of model parameters (P) and
Akaike Information Criterion differences (AAIC) for distributional models
tested. Winning models have AAIC zero. Models with goodness-of-fit nearly
equal to winners are underscored (AAIC <3.0). 4.0 some support for specified model. AAIC >10.0 indicates no support (Burnham
and Anderson 1998). Distances between landmarks for first univariate size
FA, aphids LM 1-2, bees LM 1-4 and long-legged fly LM 3-6 and for second
univariate FA aphids LM 2-3, bees LM 2-6 and long-legged fly LM 4-5.


Multivariate shape FA Multivariate size FA First univariate size FA


MLL
255.117
264.72
256.723
734.726
2241
3523
637.044


132.908
130.224
136.361
523.667
3371
5319
1148


146.603
128.617
149.585
506.553
2985
5147
1041


A AIC
0.000
17.207
1.213
955.218
3968
6530
761.855


7.369
0.000
12.275
784.888
6480
10370
2036


37.971
0.000
41.935
753.871
5710
10030
1825


MLL
542.906
547.44
543.288
1261
2847
2370
1638


536.278
549.641
535.906
1262
2180
1775
1387


452.728
454.056
455.818
1055
1760
1084
940.241


A AIC
1.236
8.303
0.000
1433
4606
3650
2189


2.744
27.47
0.000
1451
3286
2475
1703


0.000
0.656
4.18
1201
2610
1256
973.02


MLL
545.725
549.789
545.317
1265
3352
2893
1883


454.337
460.51
455.483
1081
1709
2687
1162


453.958
460.261
456.237
1064
1806
1252
1016


A AIC
2.815
8.944
0.000
1438
5612
4691
2676


0.000
10.346
0.292
1249
2505
1771
1413


0.000
10.606
2.559
1216
2701
1590
1121


Second univariate size FA

MLL AAIC
614.123 2.333
617.498 7.083
613.957 0.000
1413 1596
3534 5838
3047 4862
1950 2671


Species/model
Cotton Aphid
DPLN
RPLN
LPLN
LNORM
NORM
HNORM
LAPLACE
Honey Bee
DPLN
RPLN
LPLN
LNORM
NORM
HNORM
LAPLACE
Long-legged Fly
DPLN
RPLN
LPLN
LNORM
NORM
HNORM
LAPLACE


6.329
23.82
0.000
1393
3574
2816
1863


0.000
20.182
0.212
1144
2508
1368
1169


For multivariate shape analysis, right tail power-law exponents (a) were very high


(thousands), left-tail exponents (3) from 3.9-9.9, while the dispersion parameter (') was


narrowly distributed from 0.310 to 0.356. Thus, shape FA exhibited little scaling in tails


(i.e., nearly lognormal). For size-based FA, dispersion was much larger (0.57-0.74), and


power-law exponents more variable for univariate and multivariate size-based FA (left


515.93
525.675
513.765
1211
2302
1924
1445


412.007
422.993
412.902
985.894
1668
1099
997.28










tail (P) and right tail (ca) were generally low, indicating moderate power-law scaling in

both tails) as shown in Table 2-2.

Table 2-2. Best fit parameters for models in Table 1. Parameters for univariate size FA
are very similar to multivariate size FA and are not shown. Skew indexes for
asymmetric Laplace are also not shown.


Multivariate shape FA


Multivariate size FA


Cotton Location Dispersion/ Right tail


Aphid
DPLN
RPLN
LPLN
LNORM
NORM
HNORM
LAPLACE
Honey Bee


DPLN
RPLN
LPLN


-2.62
-3.01


Shape
0.356
0.415


-2.62 0.353


-2.87
0.062
0.009
0.049


1160
7.03


Left
Tail
4.04
00


0o 3.90


0.434
0.027
0.059
0.058


-3.74 0.310 8380 9.80
-4.02 0.274 5.52 oo


-3.71 0.305


LNORM -3.84
NORM 0.023
HNORM 0.007
LAPLACE 0.021
Long-legged Fly
DPLN Q


RPLN
LPLN

LNORM
NORM
HNORM
LAPLACE


00 7.75


0.327
0.008
0.018
0.018


flnA AQnf -an ICW~


2
-4.28 0.231 3.73 oo


-3.92 0.341


-4.02
0.019
0.007
0.017


0.351
0.008
0.015
0.017


oo 9.91
2


Location Dispersion/ Right tail


1.45
1.07
1.73
1.28
4.92
0.163
0.657


1.32
0.571
1.52
0.777
2.95
0.078
0.550


Shape
0.735
0.800
0.699
0.824
3.94
6.31
2.19


6.24
4.58


Left
Tail
3.01
00


0o 2.23


0.565 10.4 1.57
0.829 4.82 oo


0.510
0.850
2.14
3.64
2.24


oo 1.35


-0.067 0.683 3.09 5.14


-0.302 0.698 2.75


0.319 0.743


- 0.060
- 2.00
- 0.075
- 0.346


oo 3.85


0.783
2.24
2.11
1.06


Figure 2-5 shows the distribution of FA sample sizes from 229 studies published in

three recent meta-analyses (Polak et al. 2003; Thornhill and Moller 1998; Vollestad et al.


1999).














A. Cc tton Aphid





.J. I


B. Domestic Honeybee


, DPLN DPLN

0 005 1 0 15 02 025
obsDATA

o Lognormal 9
0xLGN 2
o 1
S005 01 0 15 02 025









12 0 002 004 006 022 008 24
10 DPLNo
x pDPLN


ob-DATA


Lognormal
X..


Figure 2-2. Distribution of multivariate shape FA of A) cotton aphid (Aphis gossipyii) B)
domestic honeybee (Apis mellifera) and C) long-legged fly (Chrysosoma
crinitus). Best fitting lognormal (dashed line, lower inset), and double Pareto
lognormal (solid line, upper inset) are indicated.


Il*lr.mr:,~ --n- I I I "



















































Figure 2-3. Distribution of multivariate centroid size FA of A) cotton aphid (Aphis
gossipyii) B) domestic honeybee (Apis mellifera) and C) long-legged fly
(Chrysosoma crinitus). Best fitting half-normal (dashed line, lower inset) and
double Pareto lognormal distribution (solid line, upper inset) are indicated.












A Cotton Aphid (LM 2-3)


DPLN

o o 0 f-normal
oboATA



Half-normal
, i i i~


B Domestic Honeybee (LM 1-4)


B Domestic Honeybee (LM 2-6)1
xpDaFN


DPLN

obr ATA

Half-normal ; +
n5, i


Fly (LM 3-6) l
S,- DPLN




Half-normal

5 I


C Long-legged Fly (LM 4-5)


Nf


A Cotton Aphid (LM 1-2)


Figure 2-4. Distribution of univariate unsigned size FA of A) cotton aphid (Aphis
gossipyii) B) domestic honeybee (Apis mellifera) and C) long-legged fly
(Chrysosoma crinitus). Best fitting half-normal (dashed line, lower inset) and
double Pareto lognormal distribution (solid line, upper inset) are indicated.


'\1 h P


20 DPLN

o o 0 o o 4'o
obhDATA



Half-normal








0 5 0 10
DPLN




obsDATA
Half-normal











DPLN


Hfr
obDATA

Half-normal
i i i










40-



30



20



10



0


sample or treatment size

Figure 2-5. Distribution of sample sizes (n) from 229 fluctuating asymmetry studies
reported in three recent meta-analyses (Vollestad et al. 1999, Thornhill and
Moller 1998 and Polak et al. 2003). Only five studies had sample sizes greater
than 500 (not shown).

Nearly 70% of the 229 FA studies have sample or treatment sizes less than 100.

Figure 2-6 demonstrates the hypothetical error levels (coefficients of variations) in

estimated mean FA at various sample sizes. Approximate best fit parameters were used

to estimate the coefficient of variation (CV) under the DPLN distribution (for shape FA,

v = -3.7, T = 0.35, a = 1000, 0 = 9; for size FA, v = 1.2, T = 0.70, a = 4.0, 0 = 4.0). For

the same set of landmarks, multivariate shape FA measures lead to the least amount of

error in estimating mean FA under DPLN at any sample size. Both univariate and

multivariate size FA perform more poorly in terms of both convergence and overall

percentage error.

I found that while phenodeviants occurred in almost all regions of the distribution

range of FA, the percentage of phenodeviant individuals increased dramatically with










increasing FA (Figure 2-7; aphids, r = 0.625 p = 0.013; bees, r = 0.843 p = 0.001). I also

found that individuals with phenodeviant wings (both aphids and bees) showed

significantly higher levels of FA across those wing landmarks unaffected by the

phenodeviant traits (p < 0.002 in both aphids and bees). Only a single case of

phenodeviance was sampled in long-legged flies.

45
4 0 1 .. ...............
40
35
3 0 ...................... .. .. .. .... .... .. ..............................................
o30 Legend
0 ............................................................................... ................. heg epn d F ....................... ... .............

% ERROR IN ESTIMATION 25 shape FA
OF THE MEAN (CV) ----- size FA
20
2 0 .. ... .. ... .. ... .... .... ..... ... ... .... ... .... .... ... ... .... ................................................ ....... .....................

15

0
0 \ \. I, '. i ii

0 50 100 150 200 250 300 350 400 450 500

SAMPLE SIZE
Figure 2-6. Relationship between sample size and % error for estimates of mean FA
drawn from best fitting size (dashed line) and shape (solid line) distributions
using 1000 draws per sample size. All runs use typical winning double Pareto
lognormal parameters (shape FA v = -3.7, T = 0.2, a = 1000, 0 = 9; for size FA
v = 1.2, z = 0.7, a = 4.0, 0 = 4.0).

Percent measurement error for shape FA was 1.41% in aphids, 1.63% in bees, and

2.42 % in flies while for size FA, it was -4.5% in aphids, -5% in bees, and -6.5 % in

flies. In a Procrustes ANOVA (Klingenberg and McIntyre 1998) the mean squares for

the interaction term of the ANOVA (MSinteraction) was highly significant p<0.001 in all

three species indicating that FA variation was significantly larger than variation in

measurement error (ME). The distribution of signed ME was normal and exhibited











moderate platykurtosis for all types of FA in all species examined. Measurement error


was very weakly correlated to FA in all samples (0.01 < r2 <0.07).

300 300
A B
250 250 -

200 200
Frequency
150 150

100 100
Legend IIIILegend
50 n phenodeviants I phenodeviants
50 normal 50 normal

0 0
&'' '' & .O k '',:' .... ...... O *' o g"O" C o" O
o o. r t $ *0
Procrustes Distance Procrustes Distance
064 030

C 0 D
So 028
o60
Q 027
058


054 024

S052 023
NORMALWING PHENODEVIANT NORMALWING PHENODEVIANT
APHID WINGS BEE WINGS


Figure 2-7. The proportion and percentage (inset) of individuals with visible
developmental errors on wings (phenodeviants) are shown for cotton aphids
(A) and honeybees (B) in relation to distribution of shape FA (Procrustes
distance). Average FA for both normal and phenodeviant aphids (C) and
honeybees (D) are also given.


Discussion


The Distribution of FA


The data demonstrate a common pattern of distribution in the FA in wing size and


shape of three different species whose populations existed under very different


environmental conditions (lab culture, free-living domesticated, and wild) and genetic


structure (monoclonal, haplodiploid sisters, and unrelated). The similarities across the


very different species and rearing conditions used in this study suggest that the









distribution of size and shape FA may have universal parameters (e.g., T 0.35 for shape

FA and T 0.7 for size based FA). The data confirm that although size FA sometimes

exhibits reasonable fit to half-normal in the upper tail, and shape FA is reasonably well fit

by lognormal distributions, large datasets of FA in both size and shape are always best

described by a double Pareto lognormal distribution (DPLN) or one of its limiting forms,

LPLN and RPLN. Multivariate shape FA demonstrates narrow distribution with a large

right tail, including the top few percent of the most extremely asymmetric individuals,

that is best fit by DPLN or RPLN. Both univariate and multivariate size FA exhibit a

considerably broader distribution with moderate leptokurtosis that is best fit by DPLN or

LPLN. The data suggest that the DPLN distribution and its limiting forms are generally

the most appropriate models for the distribution of FA regardless of method of

measurement.

Evidence that distribution of FA closely follows DPLN, a continuous mixture

model, and appearance of phenodeviance across nearly the entire range of data suggested

that developmental errors may be caused by a similar process across the entire

distribution of FA in a population. In other words, variation in FA may have a single

cause in most of the data. Although phenodeviance is significantly related to increased

levels of FA and is more prevalent in the right tail region of the shape FA distribution, it

does not appear associated exclusively with the right tail, as a threshold model for high

FA might predict. The very poor fits to all variations on the discrete mixture model also

suggest a lack of distinct processes creating extreme FA in the three datasets. However, I

caution that use of maximum likelihood methods to fit data to discrete mixtures is often

technically challenging. I found no block effects (e.g., no differences in FA levels









between long-legged fly samples collected on different weeks or aphid samples collected

from different pots or growth chambers), so it appears there is no obvious sample

heterogeneity that could result in a discrete mixture. With the usual caveats about

inferring process from pattern, I do not find obvious thresholds in the distribution of

asymmetries at the population level that would suggest threshold effects at a genetic or

molecular level. Lastly, based on the appearance of only one phenodeviant among our

wild trapped long-legged fly population (as opposed to many in the bee and lab reared

aphid populations), I speculate that mortality related to phenodeviance (and perhaps high

FA) in wing morphology may be relaxed in lab culture and domestication. But this could

be confounded by other differences between the three datasets including genetic

redundancy and species differences. Further comparisons among populations of single

species under different conditions would be needed to test this idea.

Sample Size and the Estimation of Mean FA

In random sampling under DPLN, we found that broad distribution effects due to

the shape parameter were minimal in their effect of slowing convergence to the

population mean in multivariate shape-based FA ('z 0.35 for all three datasets).

However, these effects are considerable for univariate and multivariate size-based FA

(where T Z 0.70). The effects of scaling in the tails of the distribution, which cause

divergence from the mean, appear to have little effect in the right tail of the distribution

of shape-based FA. However, larger effects in the tails of the size FA create more

individuals with very low and very high asymmetry than expected under the assumption

of normality. The point estimates of the scaling exponents for size FA are close to the

range where very extreme values may be sampled under the distribution tails (if a < 3 or

p <3), greatly affecting confidence in the estimate of mean FA. With a sample size of 50









and the best fitting DPLN parameters typical for asymmetry in multivariate shape, we

found that the coefficient of variation for mean FA is about 5%, whereas size-based mean

FA fluctuates about 13% from sample to sample. At a sample size of 100, these

coefficients of variation are 3.2% and 7.5% respectively. Unless experimental treatment

effects in most FA studies are larger than this, which is unlikely in studies using size-

based measures of FA of more canalized traits, statistical power and repeatability will be

low. Given the sample size range of most previous studies (n = 30-100) and their

tendency to favor size-based measurement methods, our results suggest that many past

FA studies may be under-sampled. Furthermore, it is also likely that given the small

sample sizes in many FA studies, particularly involving vertebrates where n < 50, the tail

regions of natural FA distributions are often severely under sampled and sometimes

truncated by the removal of outliers. These factors may artificially cause non-normal

distributions to appear normal, also potentially resulting in inaccurate estimation of mean

FA.

The Basis of Fluctuating Asymmetry

The surprisingly good fit of FA distributions to the DPLN model in our study

suggests that the physical basis of FA may be created by the combination of random

effects in geometrically expanding populations of cells on either side of the axis of

symmetry (i.e., geometric Brownian motion). Studies in the Drosophila wing indicate

that cell lines generally compete to fill a prescribed space during development with more

rapidly dividing lines out-competing weaker ones (Day and Lawrence 2000). Because

regulation of the growth of such cell populations involves either nutrients and/or

signaling substances that stop the cell cycle when exhausted, it is likely that the

distribution of numbers of cells present at the completion of growth follows an









exponential distribution. Reed and Jorgensen (2004) demonstrate that when a population

of repeated geometric random walks is "killed" at such a constant rate, the DPLN

distribution is the natural result. There are many other examples of growth processes in

econometrics and physics where random proportional change combined with random

stopping/observation create size distributions of the kind described here (Reed 2001;

Kozubowski and Podgorski 2002). In the future, when applying this model to instability

during biological growth, it would be very interesting to investigate how genetic and

environmental stress might affect the parameters of this model. If scaling effects are

found to vary with stress, then leptokurtosis may potentially be a better candidate signal

of developmental instability than increased mean FA.

Conclusion

Although size-based FA distributions can sometimes appear to fit normal

distributions reasonably well as previous definitions of FA suppose, I demonstrate that

three large empirical datasets all support a new statistical model for the distribution of FA

(the double Pareto lognormal distribution), which potentially exhibits power-law scaling

in the tail regions and leading to uncertain estimation of true population mean at sample

sizes reported by most FA studies. The assumption of normality fails every time

candidate models are compared on large datasets. Failure of this assumption in many

datasets may have been a major source of discontinuity in results of past FA studies.

Future work should attempt to collect larger sample/treatment sizes (n = 500) unless the

magnitude of treatment effects on FA (and thus the statistical power of comparisons) is

very large. Our results demonstrate that multivariate shape-based methods (Klingenberg

and McIntyre 1998) result in more repeatable estimates of mean FA than either

multivariate or univariate size-based methods. I would also recommend that









methodology be re-examined even in large sample studies of FA. For example, because

Drosophila are usually reared in many replicates of small tubes with less than 50 larvae

per tube, many large studies may still be compromised by individual sizes of replicate

samples. I also suggest that authors of past meta-analyses and reviews of FA literature

reassess their conclusions after excluding studies in which under-sampling is found to be

problematic. Careful attention to distributional and sampling issues in FA studies has the

potential both to mitigate problems with repeatability and possibly to suggest some of the

underlying mechanisms driving variation in FA among individuals, populations, and

species.














CHAPTER 3
INBREEDING REDUCES POWER-LAW SCALING IN THE DISTRIBUTION OF
FLUCTUATING ASYMMETRY: AN EXPLANATION OF THE BASIS OF
DEVELOPMENTAL INSTABILITY

Introduction

Fluctuating asymmetry (FA) is the average difference in size or shape of paired or

bilaterally symmetric morphological trait sampled across a population. The study of FA,

thought to be a measure of developmental instability, has a controversial history.

Fluctuating asymmetry is hypothesized by some to universally represent a population's

response to environmental and/or genetic stress (Parsons 1992, Clarke 1993, Graham

1992). It is also generally accepted that FA may be co-opted as an indicator or even a

signal of individual genetic buffering capacity to environmental stress (Moller 1990,

Moller and Pomiankowski 1993). Recent literature reviews reveal that these conclusions

are perhaps premature and analyses of individual studies often demonstrate conflicting

results (Lens 2002, Bjorksten 2000). Babbitt et al. (2006) demonstrate that this conflict

may be caused by under-sampling due to a false assumption that FA always exhibits a

normal distribution. Also, FA may be responding to experimental treatment in a complex

and as yet unpredictable fashion. Until the basis of FA is better understood, general

interpretation of FA studies remains difficult.

What is the Basis of FA?

Earlier studies of sexual selection and FA conclude that FA is ultimately a result

of strong selection against the regulation of the development of a particular morphology

(e.g., morphology used in sexual display). Thus in some instances FA may increase and









therefore become a better signal and/or more honest indicator of good genes. Sexually

selected traits tend to have increased FA (Moller and Swaddle 1997), however the exact

mechanism by which FA increases remains unexplained (i.e., a black box). More

recently, theoretical attempts have been made to explain how FA may be generated but

none have been explicitly tested. Models for the phenomenological basis of FA fall into

two general categories: reaction-diffusion models and diffusion-threshold models

(reviewed in Klingenberg 2003). The former class of models involves the chaotic and

nonlinear dynamics in the regulation or negative feedback among neighboring cells

(Emlen et al. 1993) or adjacent bilateral morphology (Graham et al. 1993). The latter

class of models combines morphogen diffusion and a threshold response, the parameters

of which are controlled by hypothetical genes and a small amount of random

developmental noise (Klingenberg and Nijhout 1999). The result of this latter class of

model is that different genotypes respond differently to the same amount of noise,

providing an explanation for genetic variation in FA response to the same environments.

Traditionally, models for the basis of FA assume that variation in FA arises from

independent stochastic events that influence the regulation of growth through negative

feedback rather than processes that may fuel or promote growth. None of these models

explicitly or mathematically address the effect of stochastic behavior in cell cycling on

the exponential growth curve. More recently Graham et al. (2003) makes a compelling

argument that fluctuating asymmetry often results from multiplicative errors during

growth. This is consistent with one particular detail about how cells behave during

growth. For several decades there has been evidence that during development, cells

actually compete to fill prescribed space until limiting nutrients or growth signals are









depleted (Diaz and Moreno 2005, Day and Lawrence 2000). Cell populations effectively

double each generation until signaled or forced to stop. It has also been observed that in

Drosophila wing disc development, that synchrony in cell cycling does not occur across

large tissue fields but rather extends only to an average cluster of 4-8 neighboring cells

regardless of the size and stage of development of the imaginal disc (Milan et al. 1995).

The assumption of previous models, particularly the reaction-diffusion type, that cell

populations are collectively controlling their cell cycling rates across a whole

developmental compartment is probably unrealistic. Regulatory control of fluctuating

asymmetry almost certainly does occur, but probably at a higher level involving multiple

developmental compartments where competing cells are prevented from crossing

boundaries. However, given that individual cells are behaving more or less

autonomously during growth within a single developmental compartment, I suggest that

variation in fluctuating asymmetry can be easily generated at this level by a process

related to stochastic exponential expansion and its termination in addition to regulatory

interactions that probably act at higher levels in the organism. In this paper, I explore and

test simple model predictions regarding the generation of fluctuating asymmetry though

multiplicative error without regulatory feedback.

Exponential Growth and Non-Normal Distribution of FA

In previous work, Babbitt et al. (2006) demonstrate that the distribution of unsigned

FA best fits a lognormal distribution with scaled or power-law tails (double Pareto

lognormal distribution or DPLN). This distribution can be generated by random

proportional (exponential) growth (or geometric Brownian motion) that is stopped or

observed randomly according to a negative exponential probability (Reed and Jorgensen

2004). I suggest that this source of power-law scaling in the tails of the FA distribution is









also the cause of leptokurtosis that is often observed empirically in the distribution of FA.

Kurtosis is the value of the standardized fourth central moment. Like the other moments,

(location, scale, and skewness), kurtosis is best viewed as a concept that can be

formalized in multiple ways (Mosteller and Tukey 1977). Leptokurtosis is best

visualized as the location and scale-free movement of probability mass from the

shoulders of a symmetric distribution towards both its center and tail (Balanda and

MacGillivray 1988). Both the Pareto and power-function distributions have shapes

characterized by the power-law and a large tail and therefore exhibit a lack of

characteristic scale. Both because kurtosis is strongly affected by tail behavior, and

because leptokurtosis involves a diminishing of characteristic scale in the shape of a

distribution, the concepts of scaling and kurtosis in real data can be, but are not

necessarily always, inter-related.

Both in the past and very recently, leptokurtosis in the distribution of FA has been

attributed to a mixture of normal FA distributions caused by a mixing of individuals, all

with different genetically-based developmental buffering capacity, or in other words,

different propensity for expressing FA (Gangestad and Thornhill 1999, Palmer and

Strobek 2003, Van Dongen et al. 2005). Although not noted by these authors, continuous

mixtures of normal distributions generate the Laplace distribution (Kotz et al. 2001,

Kozubowski and Podgorski 2001) and can be distinguished from other potential

candidate distributions by using appropriate model selection techniques, such as the

Akaike Information Criterion technique (Burnham and Anderson 1998). Graham et al.

(2003) rejects the typical explanation of leptokurtosis through the mixing of normal

distributions by noting that differences in random lognormal variable can generate









leptokurtosis. Babbitt et al. (2006) also reject the explanation that leptokurtosis in the

distribution of FA is caused by a mixture of normal distributions because the double

Pareto-lognormal distribution, not the Laplace distribution, always appears the better fit

to large samples of FA. Therefore, leptokurtosis, often observed in the distribution of

FA, may not be due to a mixing process, but instead may be an artifact of scaling in the

distribution tail, which provides evidence of geometric Brownian motion during

exponential expansion of populations of cells.

Testing a Model for the Basis of FA

I propose that the proximate basis for variation of FA in a population of organisms

is due to the random termination of stochastic geometric growth. The combination of

opposing stochastic exponential functions results in the slow power-law decay that

describes the shape of the distribution's tail. In this paper, I present a model for FA and

through simulation, test the prediction that genetic variation in the ability to precisely

terminate growth will lead to increased kurtosis and decreased scaling exponent in the

upper tail of the distribution. Then, I assess the validity of this model by direct

comparison to the distribution of FA within large samples of wild and inbred populations

of Drosophila. Under the assumption that a less heterozygous population will have less

variance in the termination of growth, I would predict that inbreeding should act to

reduce power-law scaling effects in the distribution of FA in a population. Inbreeding

should also reduce the tail weight kurtosiss) and mean FA assuming inbred individuals

have lower variance in the times at which they terminate growth. I also assume that

inbreeding within specific lines does not act to amplify FA due to inbreeding depression.

It has been demonstrated that Drosophila melanogaster do not increase mean FA in

response to inbreeding (Fowler and Whitlock 1994) and it is suggested that large









panmictic populations typical of Drosophila melanogaster may not harbor as many

hidden deleterious recessive mutations as other species (Houle 1989), making them

resistant to much of the typical genetic stress of inbreeding. Therefore, the absence of

specific gene effects during inbreeding suggests that Drosophila may be a good model

for investigating the validity of our model as an explanation of the natural variation

occurring in population level FA.

Methods

Model Development

Simulation of geometric Brownian motion

Ordinary Brownian motion is most easily simulated by summing independent

Gaussian distributed random numbers or white noise (X,). See Figure 3-1.


(3.1) W(X)= X,
i=1

which simulated in discrete steps is


(3.2) N,= Nt, + 1

where N = cell population size, t = time step and W = a random Gaussian variable.

Exponential or geometric Brownian motion, a random walk on a natural log scale, can be

similarly simulated. Geometric Brownian motion is described by the stochastic

differential equation

(3.3) dY(t) = pY(t)dt + vY(t)dW(t)

or also as


(3.4) d =(t) udt + udW(t)
Y(t)










WHITE NOISE










ORDINARY BROWNIAN MOTION
-10050 If----------------------------

1000

600

N 200

47 70 93 116139162185 08 231 2542773003233466 43
-400
-600

time

Figure 3-1. Ordinary Brownian motion (lower panel) in N simulated by summing
independent uniform random variables (W) (upper panel).

where W(t) is a Brownian motion (or Weiner process) and / and v are constants that

represent drift and volatility respectively. Equation 3.3 has a lognormal analytic solution


(3.5) Y(t) = Y(O)e(-' 12)t+vW(t)

A simulation of geometric Brownian motion in discrete form follows as


(3.6) Nt = Nt-,1 t-W,


where N = cell population size, t = time step and W = a random Gaussian variable. See

Figure 3-2. Equation 3.6 is identical to the equation for multiplicative error in Graham et

al. (2003). I modify equation 3.6 slightly by letting Wrange uniformly from 0.0 to 1.0


with W= 0.5 and adding the drift constant C that allows for stochastic upward drift (at C

> 0.5) or downward drift (at C < 0.5).












(3.7) N -= CN1 + Nt-lW


= (C + W,)N,,


At C = 0.5, eqns. 3.6 and 3.7 behave identically. Geometric Brownian motion with


upward drift is shown in Figure 3-3.


0.00045

0.0004

0.00035

0.0003

0.00025

0.0002

0.00015

0.0001

0.00005

0


35 69 103 137 171 205 239 273 307 341 375 409 443 477
time step


Log N


1
0.1
0.01
0.001
0.0001
0.00001
0.000001
0.0000001


time step


time


Figure 3-2. Geometric Brownian motion in N and log N simulated by multiplying
independent uniform random variables. This was generated using Equation
1.5 with C = 0.54.


36 71 106 141 176211 246 281 316 351 386421 456


j___A__ War_
AA VW


-
-
-







44



4500000000
4000000000
3500000000
3000000000
2500000000
N 2000000000
N
1500000000
1000000000
500000000
0
1 37 73 109 145 181 217 253 289 325 361 397 433 469
time step

1E+11
1E+09
1000000
0
100000

Log N 1000
10 -
01 34 133 166 199 232 265 298 331 364 397 430 463
0 001
000001
time step


time


Figure 3-3. Geometric Brownian motion in N and log N with upward drift. This was
generated using Equation 1.5 with C = 0.60.



Simulation of fluctuating asymmetry

Using the MathCad 13 (Mathsoft Engineering and Education 2005), two


independent geometric random walks were performed and stopped randomly at mean


time t= 200 steps with some variable normal probability. The random walks result in cell


population size equal to N, (orNf and NI on left and right sides respectively).


Fluctuating asymmetry (FA) was defined as the difference in size resulting from this


random proportional growth on both sides of the bodies axis of symmetry plus a small


degree of random uniformly distributed noise or










(3.8) FA= NL -N +rv

where (rv) was uniformly distributed with a range of +0.1 ( N N ). Using a

MathCad-based simulation in VisSim LE, the generation of individual FA values was

repeated until a sample size of 5000 was reached. The random noise (rv) has no effect on

the shape of the FA distribution but fills empty bins (gaps) in distribution tails. Because

rv is small in comparison to N NJ, its effect is similar to that of measurement error

(which would be normally distributed rather than uniform). Schematic representation of

the simulation process for Reed and Jorgensen (2004) model and simulation of

fluctuating asymmetry are shown in Figure 4 and 5. The simulation of the distribution of

fluctuating asymmetry was compared at normal standard deviation of termination of

growth (t) ranging from a = 0.5, 0.8, 1.2,3 and 7 with a drift constant ofC = 0.7.

Inbreeding Experiment

In May 2004 in Gainesville, Florida, 320 free-living Drosophila simulans were

collected in banana baited traps and put into a large glass jar and cultured on instant rice

meal and brewer's yeast. After two generational cycles 1000 individuals were collected

in alcohol. This was repeated again in June 2005 with 200 wild trapped flies.

Lines of inbred flies were created from the May 2004 wild population through eight

generations of full sib crosses removing an estimated 75% of the preexisting

heterozygosity (after Crow and Kimura 1970). Initially, ten individual pairs were

isolated from the stock culture and mated in 1/2 pint mason jars with media and capped

with coffee filters. In each generation, and in each line, and to ensure that inbred lines

were not accidentally lost though an inviable pairing, four pairs of Fl sibs from each

cross were then mated in 12 pint jars. Offspring from one of these four crosses were











randomly selected to set up the next generation. Of the original ten lines, only four

remained viable after eight generations of full sib crossing. These remaining lines were

allowed to increase to 1000+ individuals in 1 quart mason jars and then were collected

for analysis in 85% ethanol. This generally took about 4 generations (8 weeks) of open

breeding. One completely isogenic (balancer) line of Drosophila melanogaster was

obtained from Dr. Marta Wayne, University of Florida and also propagated and collected.
















0 10 20 30 40 50 60 70 80 90 100
I001










1 DOUBLE PARETO LOGNORMAL DISTRIBUTIONITH











NEGATIVE EXPONENTI AL REPEAT
VARIATION IN GROWTH SAMPLING
0 10 20 30 40 50 60 70 80 90 100













STOP TIME
T SO DOUBLE PARETO LOGNORMAL DISTRIBUTION













'0 100
o VARIATION IN GO 00










Figure 3-4. Model representation of Reed and Jorgensen's (2004) physical size
distribution model. Variable negative exponentially distributed stopping
Times of random proportional growth (GBM with C = 0.5) create double
Pareto lognormal distribution of size.












Stochastic Geometric Growth


RANDOM WALKS WITH
GEOMETRIC BROWNIAN
MOTION


GROWTH STOP TIME


SIZE DISTRIBUTION


I_ \ 1

REPEAT
NORMAL VARIATION SAMPLING
IN GROWTH STOP TIME





0 50 100 150

Figure 3-5. A model representation of developmental instability. Normally variable
stopping times of random proportional growth (GBM with C > 0.5) create
double Pareto lognormal distribution of size.


All flies were cultured on rice meal and yeast at 290C with 12:12 LD cycle in

environmentally controlled chambers at the Department of Entomology and Nematology

at the University of Florida. Wings were collected and removed from 1000 flies from

each of the two samples of wild population and four samples of inbred lines and dry

mounted on microscope slides. Specimens were dried in 85% ethanol, and then pairs of









wings were dissected (in ethanol) and air-dried to the glass slides. Permount was used to

attach cover slips. This technique prevented wings from floating up during mounting,

which might slightly distort the landmark configuration. Dry mounts were digitally

photographed. All landmarks were identified as wing vein intersections on the digital

images (eight landmarks on each wing). See Appendix A for landmark locations.

Morphometric analyses

Wing vein intersections were digitized on all specimens using TPSDIG version

1.31 (Rohlf, 1999). Specimens damaged at or near any landmarks were discarded.

Fluctuating asymmetry was measured in two ways on all specimens using landmark-

based multivariate geometric morphometrics. A multivariate size-based FA (FA 1 in

Palmer and Strobek 2003) was calculated as absolute value of(R L) or just R-L in

signed FA distributions where R and L are the centroid sizes of each wing (i.e., the sum

of the distances of each landmark to their combined center of mass or centroid location).

In addition, a multivariate shape-based measure of FA known as the Procrustes distance

was calculated as the square root of the sum of all squared Euclidean distances between

each left and right landmark after two-dimensional Procrustes fitting of the data

(Bookstein 1991; Klingenberg and McIntyre 1998; FA 18 in Palmer and Strobeck 2003;

Smith et al. 1997). Centroid size calculation, Euclidean distance calculation and

Procrustes fitting were performed using 0yvind Hammer's Paleontological Statistics

program PAST version 0.98 (Hammer 2002). A sub-sample of 50 individuals from the

fourth inbred line (pp4B3) was digitized five times to estimate measurement error. In

these cases, measures of FA were taken as the average FA value of the five replicate

measurements for each specimen. Percent measurement error was also computed as









(ME/average FA) x 100 whereME = sd(FA, FA2, FA3, FA4, FA5) (as per Palmer and

Strobek 2003). For assessing whether measurement error (ME) interfered significantly

with FA, a Procrustes ANOVA (in Microsoft Excel) was performed on the five

replications of the 50 specimen sub-sample (Klingenberg and McIntyre 1998). Any

subsequent statistical analyses were performed using SPSS Base 8.0 statistical software

(SPSS Inc.).

Model selection and inference

The fits of unsigned size FA to three distributional models (half-normal, lognormal,

and double Pareto lognormal (DPLN)) were compared in the Drosophila lines, by

calculating negative log likelihood and Akaike Information Criteria (AIC) (Burnham

and Anderson 1998; Hilborn and Mangel 1998). This method penalizes more complex

models (those with more parameters) when selecting the best-fit distributional model for

a given dataset. Best fitting parameters were obtained by maximizing the log-likelihood

function for each model (Appendix B). The maximization was performed using the

conjugate gradient method within unconstrained solve blocks in the program MathCAD

by MathSoft Engineering and Education Inc (2001).

Results

Model Simulation

The amount of variance in termination times related directly to levels of simulated

FA (i.e., low variance in termination time (t) gives low FA and vice versa). I found that

not only does amount of FA increase with increased variance in (t), but so do both

kurtosis and the scaling effect in the distribution tails. In Figure 3-6, normal quantile

plots of signed FA are shown for different standard deviations in the normal variation of

the termination of growth of geometrically expanding cell populations. The degree of the







50


S or sigmoidal shape in the plot indicates level ofleptokurtosis. The leptokurtosis in the

quantile plot is reduced greatly with a decrease in the standard deviation of the normal

variation in growth termination times.


S= 0.5
kurtosis


o =0.8
kurtosis


- =1.2
kurtosis


0.634







0.933






=1.149


o =3.0
kurtosis = 11.812


r = 7.0
kurtosis


:27.266


SIGNED DISTRIBUTION OF
FLUCTUATING ASYMMETRY


4
7a-t


a









fl
th


DETRENDED FIT
TO NORMAL


Figure 3-6. Simulated distributions of cell population size and FA for different amounts
of variation in the termination of growth (variance in normally distributed
growth stop time). The fit of simulated data to the normal distribution can
determined by how closely the plotted points follow the horizontal line (a
good fit is horizontal).


I


I









Experimental Results

Both mean unsigned size FA and shape FA decreased with inbreeding in all lines.

I also observed that the kurtosis of signed size FA and the skewness of both unsigned size

and shape FA follow an identical trend. The trend was strongest in kurtosis, which

decreased rapidly with inbreeding, indicating that, as predicted, changes in mean FA are

influenced strongly by the shape and tail behavior of the distribution of FA (Table 3-1).

Table 3-1. Distribution parameters and model fit for multivariate FA in two wild
populations and four inbred lines of Drosophila simulans and one isogenic
line of Drosophila melanogaster. Model fits are A AIC for unsigned centroid
size FA (zero is best fit, lowest number is next best fit).

Kurtosis Mean Mean Scaling
(signed (unsigned (shape FA) exponent AAIC AAIC AAIC
size FA) size FA) in upper HNORM LGN DPLN
tail
Wild 2004 28.3 5.73 0.0223 1.49 90.6 324 0.000
+0.12 0.38 0.0003
Wild 2005 45.9 4.68 0.0237 2.87 66.2 261 0.000
+0.15 0.19 0.0003
Inbred line 2.43 3.27 0.0183 5.09 0.000 104 15.5
1 (OR18D) 0.11 0.09 0.0002
Inbred line 8.85 4.24 0.0212 3.10 11.50 223 0.000
2(OR18D3) 0.15 0.13 0.0002
Inbred line 2.90 3.31 0.0184 4.63 0.000 130 44.0
3 (PP4B2) 0.15 0.09 0.0002
Inbred line 3.30 3.72 0.0213 5.78 0.000 228 79.7
4 (PP4B3) 0.15 0.11 0.0003
Isogenic 0.550 3.90 0.0201 9.99 0.000 84.9 101
(me175) 0.11 0.10 0.0002

While wild populations on the whole, demonstrated increased mean, skew and

leptokurtosis of FA compared to inbred lines, significant differences between populations

were found in all distributional parameters, even between each of the inbred lines

(ANOVA mean size FA for all lines; F = 27.49, p < 0.001; ANOVA mean size FA for

inbred lines only; F = 14.38, p < 0.001; note shape FA also show same result). Overall,









inbred lines demonstrated lower kurtosis, just slightly above that expected from a normal

distribution (Table 1). They also had lower mean FA. No significant differences were

found with respect to these results according to sexes of flies. In Figure 3-7, I show the

distribution of FA and detrended normal quantile plots for the wild population, four

inbred lines and one isogenic line respectively.

As in the simulated data, the degree of the S shape in the plot indicated level of

leptokurtosis. The S shape in quantile plot is reduced greatly with inbreeding and nearly

disappears in the isogenic line.

Model Selection and Inference

The comparison of candidate distributional models of FA demonstrated normalization

associated with inbreeding in three of the four inbred lines (Table 3-1). In the remaining

inbred line, the half-normal candidate model was a close second to the double Pareto

lognormal distribution. In the wild population samples, the distribution of unsigned size-

based FA was best described by the double Pareto lognormal distribution (DPLN), a

lognormal distribution with scaling in both tails. In the best fitting parameters of this

distribution there was no observable trend in lognormal mean or variance across wild

populations and inbred lines. The scaling exponent of the lower tail (P in Reed and

Jorgensen 2004) was close to one in all lines while the scaling exponent in the upper tail

(a in Reed and Jorgensen 2004) increased with inbreeding (Table 3-1). Both samples of

the wild population demonstrated the lowest scaling exponent in the upper tail of the

distribution. The low scaling exponents here indicate divergence in variance (2004 and

2005 where a < 3) and in the mean (2004 only where a < 2). The inbred lines all show

higher scaling exponents that are consistent with converging mean and variance.













wild 2004
kurtosis = 28.25


wild 2005
kurtos =45.85





inbred -OR18D
kurtosis =2.43






inbred -OR18D3
kurtosis =8.85






inbred -PP4B2
kurtosis = 2.90


kurtosis =3.30






isogenic mel 75
kurtosis =0.55


aP


II

A




r







a
U,\


[r


eat ai


Figure 3-7. Distribution of fluctuating asymmetry and detrended fit to normal for two
samples of wild population collected in Gainesville, FL in summers of 2004
and 2005 and four inbred lines of Drosophila simulans derived from eight
generations of full-sib crossing of the wild population of 2004. Also included
is one isogenic line of Drosophila melanogaster (me175). All n = 1000. The
fit of data to the normal distribution can determined by how closely the
plotted points follow the horizontal line (a good fit is horizontal).


-a


I


~----~-
"









The isogenic Drosophila melanogaster line demonstrated the highest scaling exponent

and the most normalized distribution of unsigned FA.

Measurement error was 7.6% for shape-based FA and 13.0% for centroid size-

based FA. In a Procrustes ANOVA (Klingenberg and McIntyre 1998) the mean squares

for the interaction term of the ANOVA (MSinteraction) was highly significant p<0.001

indicating that FA variation was significantly larger than variation due to measurement

error (ME).

Discussion

Revealing the Genetic Component of FA

While we should be cautious about inferring process from pattern, the very similar

results of both the modeling and the inbreeding experiment in Drosophila seem to

suggest the presence of a scaling component in the distribution of fluctuating asymmetry

that is caused by a random multiplicative growth process as suggested previously by

Graham et al. 2003. This parameter appears to change with the genetic redundancy of the

population which is presumably increased by genetic drift and and reduction in

heterozygosity during inbreeding. Specifically, the scaling exponent(s) of the upper tail

(a) of the unsigned FA distribution, or outer tails of the signed FA distribution, are

increased with inbreeding, causing more rapid power-law decay in the shape of the tails.

This effect also reduces kurtosis and apparently normalizes the distribution of FA in more

inbred populations. This suggests that individual genetic differences in the capacity to

control variance in the termination of random proportional growth (i.e., geometric

Brownian motion) may be responsible for determining the shape and kurtosis of the

distribution of FA. In other words, leptokurtosis kurtosiss >3) in signed FA distribution









indicates genetic variability in the population while normality kurtosiss = 3) indicates

genetic redundancy.

Because leptokurtosis is very often observed in the distribution of FA, genetic

variability potentially underlies a large proportion of the variability observed in the FA of

a given population. Observed differences or changes in FA are therefore not only a

response of development to environmental stress, but clearly also can reflect inherent

differences in the genetic redundancy of populations. The significantly different levels of

mean FA among the inbred lines in this study, presumably caused by the random fixation

of certain alleles, also suggests that there is a strong genetic component to the ability to

buffer development against random noise. It is assumed that the differences observed in

FA between wild trapped and inbred populations in this study do not indicate an effect of

inbreeding depression in the study for two reasons. First, the four inbred lines analyzed

in this study were vigorous in culture so the fixation of random alleles was probably not

deleterious. Second, and more important, inbreeding reduced FA rather than increasing it

as would have been expected under genetic stress.

It is also important to note that kurtosis is potentially a much stronger indicator of

FA than the distribution mean. The low scaling exponents found in the non-normal

distributions of FA in the wild populations of Drosophila simulans are capable of

slowing and perhaps even stopping, the convergence on mean FA with increased sample

size. Because kurtosis is a fourth order moment, estimating it accurately also requires

larger sample sizes. However, if kurtosis can be demonstrated to respond as strongly to

environmental stress as it does here to inbreeding, its potential strength as a signal of FA

may allow new interpretation of past studies of FA without the collection of more data.









This may help resolve some of the current debate regarding FA as a universal indicator of

environmental health and as a potential sexual signal in "good genes" models of sexual

selection.

Limitations of the Model

There are certain aspects of the model presented here that may be

oversimplifications of the real developmental process. First of all, this model assumes

developmental instability is generated left-right growing tissue fields with no regulatory

feedback or control other than when growth is stopped. It is very likely that left-right

regulation is able to occur at higher levels of organization (e.g., across multiple

developmental compartments) even though there is no evidence of regulated cell cycling

rates beyond the distance of 6-8 cells on average within any given developmental

compartment (Milan et al. 1995). Therefore this model explains how fluctuating

asymmetry is generated, not how it is regulated. Second, this model only considers cell

proliferation as influencing size. It is known that both cell size and programmed cell

death, or apoptosis, are also important in regulating body size (Raff 1992, Conlin and

Raff 1999). Both of these may play a more prominent role in vertebrate development,

than they do in insect wings, where apparently growth is terminated during its

exponential phase. Nevertheless, this simple model seems to replicate very well, certain

behavioral aspects of the distribution of fluctuating asymmetry in the insect wing.

The Sources of Scaling

The basic process generating power-law scaling effects illustrated here (Reed 2001)

offers an alternative perspective to the often narrow explanations of power-laws caused

by self-organized criticality in the interactions among system components (Gisiger 2000).

The power law that results from self-organized criticality is created from the multiplicity









of interaction paths in the network. As the distance between two interacting objects is

increased in a network or multidimensional lattice, the number of potential interaction

pathways increases exponentially, while the correlation between such paths decreases

exponentially (Stanley 1995). These opposing exponential relationships create the power

law scaling observed in simulations of self-organized critical systems. While there is no

such "interaction" in our model, there is a power-law generated by opposing exponential

functions. The constant degree of change represented by the power law in both the

statistical physics of critical systems and the mathematics of both Reed and Jorgensen's

process and the model given here is the result of the combined battle between both the

exponentially increasing and decreasing rates of change (Reed 2001). While the natural

processes are quite different, the underlying mathematical behavior is very similar.

Potential Application to Cancer Screening

Just as changes in the shape of the distribution of fluctuating asymmetry is

normalized across a population of genetically redundant individuals, genetic redundancy

in a population of cells may also help maintain normal cell size and appearance. The loss

of genetic redundancy in a tissue is a hallmark of cancer. The abnormal gene expression

and consequent genetic instability that characterizes cancerous tissue often results in

asymmetric morphology in cells, tissues and tissue borders. Baish and Jain (2000)

review the many studies connecting fractal (scale free) geometry to the morphology of

cancer. Cancer cells also are typically pleomorphic or more variable in size and shape

than normal cells and this pleomorphy is associated with intercellular differences in the

amount of genetic material (Ruddon 1995). Frigyesyi et al. (2003) have demonstrated a

power-law distribution of chromosomal aberrations in cancer. Currently the type of

distributional shape of the pleomorphic variability in cell size is not known, or at least not









published. However, Mendes et al. (2001) demonstrated that cluster size distributions of

HN-5 (cancer) cell aggregates in culture followed a power-law scaled distribution.

Furthermore these authors also demonstrated that in MDCK (normal) cells and Hep-2

(cancer) cells, cluster size distributions transitioned from short-tailed exponential

distributions to long-tailed power-law distributions over time. The transition is

irreversible and is likely an adaptive response to high density and long permanence in

culture due to changes in either control of replication or perhaps cell signaling. Taken

collectively, these studies may suggest that scaling at higher levels of biological

organization observed in cancer is due to increased relative differences in length of cell

cycling rates of highly pleomorphic cell populations that have relatively larger

intercellular differences in amount of genetic material. The stochastic growth model I

have proposed as the basis for higher variability in population level developmental

instability or FA may also provide a possible explanation for higher variability in the cell

sizes of cancerous tissue. If genetic redundancy in growing tissue has the same

distributional effect as genetic redundancy in populations of organisms and tends to

normalize the observed statistical distribution, then one might predict that the genetic

instability of cancerous tissue would create a scaling effect that causes pleomorphy in

cells and scaling in cell cluster aggregations. Statistical comparison of cell and cell

cluster size distributions in normal and cancerous tissues may provide a useful and

general screening technique for detecting when genetic redundancy is compromised by

cancer in normal tissues.

Conclusion

Until now, the basis of fluctuating asymmetry has been addressed only with

abstract models of hypothetical cell signaling, or elsewhere, at the level of selection









working on the organism with potential mechanism remaining in the black box.

However, fluctuating asymmetry must first and foremost be envisioned as a stochastic

process occurring during tissue growth, or in other words, occurring in an exponentially

expanding population of cells. As demonstrated in the model presented here, this

expansion process can be represented by stochastic proportional (geometric) growth that

is terminated or observed randomly over time. These results imply that the fluctuating

asymmetry observed in populations is not only related to potential environmental

stressors, but also to a large degree, the underlying genetic variability in those molecular

processes that control the termination of growth. Therefore, fluctuating asymmetry

responses to stress may be hard to interpret without controlling for genetic redundancy in

the population. Both the simulation and experimental results suggest that measures of

distributional shape like kurtosis, scaling exponent and tail weight may actually be a

strong signal of variability in the underlying process that causes developmental

instability. Therefore the kurtosis parameter of the fluctuating asymmetry distribution

may provide more information about fluctuating asymmetry response than does a

populations' average or mean fluctuating asymmetry. This may provide a novel method

by which to resolve conflicts in previous under-sampled research without the collection

of more data.














CHAPTER 4
TEMPERATURE RESPONSE OF FLUCTUATING ASYMMETRY TO IN AN APHID
CLONE: A PROPOSAL FOR DETECTING SEXUAL SELECTION ON
DEVELOPMENTAL INSTABILITY

Introduction


Developmental instability is a potentially maladaptive component of individual

phenotypic variation with some unknown basis in both gene and environment (Moller

and Swaddle 1997, Fuller and Houle 2003). Developmental instability is most often

measured by the manifestation of fluctuating asymmetry (FA), the right minus left side

difference in size or shape in a single trait across the population (Palmer and Strobek

1986, 2003; Parsons 1992, Klingenberg and McIntyre 1998). Because FA is thought to

indicate stress during development, the primary interest in the study of FA has been its

potential utility as an indicator of good genes in mate choice (Moller 1990, Moller and

Pomiankowski 1993) or its utility as a general bioindicator of environmental health

(Parsons 1992).

The Genetic Basis of FA


For FA to become a sexually selected trait, it must be assumed that it has a

significant genetic basis, and can therefore evolve (Moller 1990, Moller and

Pomiankowski 1993). However, researchers who use FA as a bioindicator of

environmental health often assume that FA is a phenotypic response that is mostly

environmental in origin (Parsons 1992, Lens et al. 2002). Studies of the heritability of

FA seem to indicate low heritability for FA exists (Whitlock and Fowler 1997, Gangestad









and Thornhill 1999). However, because FA is essentially a variance that is often

measured with only two data points per individual, FA may have a stronger but less

easily detectable genetic basis (Whitlock 1996, Fuller and Houle 2003). Additive genetic

variation in FA in most studies has been found to be minimal, but several quantitative

trait loci studies suggest significant dominance and character specific epistatic influences

on FA (Leamy 2003, Leamy and Klingenberg 2005). Babbitt (chapter 3) has

demonstrated that a population's genetic variability affects the distributional shape of FA.

So while studies investigating mean FA may be inconclusive, changes in the population's

distributional shape seem to indicate potential genetic influence on FA. However, no

studies have observed FA in a clonal organism for the express purpose of assessing

developmental instability that is purely environmental (i.e., non-genetic) in its origin (i.e.,

developmental noise).

The Environmental Basis of FA


It has long been assumed that FA is the result of some level of genetically-based

buffering of additive independent molecular noise during development. Because of the

difference in scale between the size of molecules and growing cells it would be unlikely

that molecular noise would comprise an important source of variation in functioning

cells. However, Leamy and Klingenberg (2005) rationalize that molecular noise could

only scale to the level of tissue when developmentally important molecules exist in very

small quantity (e.g., DNA or protein) and therefore FA may represent a stochastic

component of gene expression. This is somewhat similar in spirit to the Emlen et al.

(1993) explanation of FA that invokes non-linear dynamics of signaling and supply that

may also occur during growth. Here FA is thought to be the result of the scaling up of









compounding temporal asymmetries in signaling between cells during growth. In this

model, hypothetical levels of signaling compounds (morphogens) and or growth

precursors used in the construction of cells vary randomly over time. When growth

suffers less interruption, thus when it occurs faster, there is also less complexity (or

fractal dimension) in the dynamics of signaling and supply. Graham et al. (1993) suggest

that nonlinear dynamics of hormonal signaling across the whole body may also play a

similarly important role in the manifestation of FA. However, while the levels of FA are

certainly influenced by the regulation of the growth process, both Graham et al. (2003)

and Babbitt (chapter 3) also suggest that FA levels reflect noise during cell cycling that is

amplified by exponentially expanding populations of growing cells.

Although the proximate basis of FA is not well understood, its ultimate

evolutionary basis, while heavily debated, is easier to understand. Moller and

Pomiankowski (1993) first suggested that strong natural or sexual selection can remove

regulatory steps controlling the symmetric development of certain traits (e.g.,

morphology used in sexual display). They suggest that with respect to these traits (and

assuming that they are somehow costly to produce), individuals may vary in their ability

to buffer against environmental stress and developmental noise in relation to the size of

their individual energetic reserves; which are in turn often indicative of individual genetic

quality. Therefore high genetic quality is associated with low FA.

Most existing proximate or growth mechanical explanations of FA assume that

rapid growth is less stressful in the sense that fewer interruptions of growth by various

types of noise should result in lower FA (Emlen et al. 1993, Graham et al. 1993).

However, ultimate evolutionary explanations of FA assume that rapid growth is









potentially more stressful because it is energetically costly and therefore rapid growth

should increase the level of FA when energy supply is limiting (Moller and

Pomiankowski 1993). This high FA is relative and so should be especially prominent in

individuals of lower genetic quality who can least afford to pay this additional energy

cost. This fundamental difference between the predictions of proximate (or mechanistic)

level and ultimate evolutionary level effects of temperature on FA is shown in Figure 1.

The theoretical difference in the correlation of temperature and growth rate to FA in both

the presence and absence of energetic limitation could be used to potentially detect sexual

selection on FA. However it first should be confirmed that FA should decrease with

more rapid growth in the absence of energetic limitation to growth and genetic

differences between individuals in a population. This later objective is the primary goal

of this study.

Temperature and FA in and Aphid Clone


At a very basic level, entropy or noise in physical and chemical systems has a

direct relationship with the physical energy present in the system. This energy is

measured by temperature. Because FA is speculated to tap into biological variation that

is somewhat free of direct genetic control, it may therefore respond to temperature in

simple ways.

First, increased temperature may increase molecular entropy which may in turn

increase developmental noise during development thereby increasing FA. Second,

increased temperature may act as a cue to shorten development time (as in many aphids

where higher temperature reduces both body size and development time), thereby

reducing the total time in which developmental errors may occur. This should reduce










FA. A third possibility is that a species specific optimal temperature exists. If so, FA

should increase while approaching both the upper and lower thermal tolerance limits of

organisms.

PROXIMATE LEVEL EFFECT


DEVELOPMENTAL +
TIME

TEMPERATURE FLUCTUATING
ASYMMETRY

GROWTH RATE


ULTIMATE LEVEL EFFECT

+
STRESS DEVELOPMENTAL
+ \^ / TIME

NON-OPTIMAL -- FLUCTUATING
TEMPERATURE ASYMMETRY
ENERGETIC G R
RESERVES GROWTH RATE
+
Figure 4-1. Predicted proximate and ultimate level correlations of temperature and
growth rate to fluctuating asymmetry are different. Ultimate level
(evolutionary) effects assume energetic limitation of individuals in the system.
Proximate level (growth mechanical) effects do not. Notice that temperature
and fluctuating asymmetry are negatively correlated in the proximate model
while in the ultimate model they are positively correlated.

Only a few studies have directly investigated the relationship between FA and

temperature. The results are conflicting. FA is either found to increase on both sides of

an "optimal" temperature (Trotta et al. 2005, Zakarov and Shchepotkin 1995), to be

highest at low temperature (Chapman and Goulson 2000), to simply increase with

increasing temperature (Savage and Hogarth 1999, Mpho et al. 2002) or not to respond









(Hogg et al. 2001). None of these studies investigate the relationship between

developmental noise (FA in a clonal line) and temperature.

The characterization of developmental noise in response to temperature was

investigated in this study using the cotton aphid, Aphis gossipyii. These aphids reproduce

parthenogenetically, are not energetically limited in their diet (because they excrete

excess water and sugar as "honeydew") and produce wings that are easily measured using

multiple landmarks. They demonstrate large visible variation in body size, wing size and

even wing FA. The visible levels of wing asymmetry in cotton aphids reflect levels of

FA that are about four times higher than that observed in other insect wings (Babbitt et al.

2006, Babbitt in press). Because parthenogenetic aphids cannot purge deleterious

mutations each generation and because Florida clones often never use sexual

reproduction to produce over-wintering eggs, this remarkably high FA may be the result

of Muller's ratchet. Cotton aphids are also phenotypically plastic in response to

temperature, producing smaller lighter morphs at high temperatures and larger dark

morphs at low temperatures. This feature allows observation of two genetically

homogeneous groups in which different gene expression patterns (causing the color

morphs) exist. The central prediction is that of the proximate effect model: that in the

absence of energetic limitation and genetic variation, temperature and environmentally

induced FA (or developmental noise) should be negatively correlated in a more or less

monotonic relationship.

Methods


In March 2003, a monoclonal population of Cotton Aphids (Aphis gossipyii

Glover) was obtained from Dr. J.P. Michaud in Lake Alfred, Florida and was brought to









the Department of Entomology and Nematology at the University of Florida. The culture

was maintained on cotton seedlings (Gossipium) grown at different temperatures (12.50C,

150C, 170C, 190C, 22.50C and 250C with n = 677 total or about 100+ per treatment)

under artificial grow lights (14L:10D cycle). Because of potential under-sampling caused

by a non-normal distribution of FA (see Babbitt et al. 2006), a second monoclonal

population collected from Gainesville, FL in June 2004 was reared similarly but in much

larger numbers at 12.50C, 150C, 170C, 190C, and 250C (n = 1677 or about 300+ per

treatment).

Development time for individual apterous cotton aphids (Lake Alfred clone) were

determined on excised cotton leaf discs using the method Kersting et al. (1999). Twenty

randomly selected females were placed upon twenty leaf discs (5 cm diameter) per

temperature treatment. Discs were set upon wet cotton wool in petri dishes and any first

instar nymphs (usually 3-5) appearing in 24 hours were then left on the discs.

Development time was taken as the average number of days taken to reach adult stage

and compared across temperatures. Presence of shed exoskeleton was used to determine

instar stages. Cotton was wetted daily and leaf discs were changed every 5 days.

Humidity was maintained at 50+5%.

In each temperature treatment, single clonal populations were allowed to increase

on plants until crowded in order to stimulate alate (winged individuals) production.

Temperature treatments above 170C produced light colored morphs that were smaller and

tended to feed on the undersides of leaves of cotton seedlings. Temperature treatments

below 17C produced larger dark morphs that tended to feed on the stems of cotton

seedlings. Alatae were collected using small brushes dipped in alcohol and stored in 80%









ethanol. Wings were dissected using fine insect mounting pins and dry mounted as pairs

on microscope slides. Species identification was by Dr. Susan Halbert at the State of

Florida Department of Plant Industry in Gainesville.

Specimens were dried in 85% ethanol, and then pairs of wings were dissected (in

ethanol) and air-dried to the glass slides while ethanol evaporated. Permount was used to

attach cover slips. This technique prevented wings from floating up during mounting,

which might slightly distort the landmark configuration. Dry mounts were digitally

photographed. Six landmarks were identified as the two wing vein intersections and four

termination points for the third subcostal. See Appendix A for landmark locations.

Wing vein intersections were digitized using TPSDIG version 1.31 (Rohlf,

1999). Specimens damaged at or near any landmarks were discarded. Fluctuating

asymmetry was calculated using a multivariate geometric morphometric landmark-based

method. All landmarks are shown in Appendix A. FA (FA 1 in Palmer and Strobek

2003) was calculated as absolute value of (R L) where R and L are the centroid sizes of

each wing (i.e., the sum of the distances of each landmark to their combined center of

mass or centroid location). In addition, a multivariate shape-based measure of FA known

as the Procrustes distance was calculated as the square root of the sum of all squared

Euclidean distances between each left and right landmark after two-dimensional

Procrustes fitting of the data (Bookstein 1991; Klingenberg and McIntyre 1998; FA 18 in

Palmer and Strobeck 2003; Smith et al. 1997). This removed any difference due to size

alone. Centroid size calculation, Euclidean distance calculation and Procrustes fitting

were performed using 0yvind Hammer's Paleontological Statistics program PAST

version 0.98 (Hammer 2002). Percent measurement error was computed as (ME/average










FA) x 100 where ME= ( FA FA2 + FA2 -FA3 + FA FA3)/3 in a smaller subset

(200 wings each measured 3 times = FA1, FA2 and FA3) of the total sample. All

subsequent statistical analyses were performed using SPSS Base 8.0 statistical software

(SPSS Inc.). Unsigned multivariate size and shape FA as well as the kurtosis of signed

FA were then compared at various temperatures using one-way ANOVA.

Results


Development time (Figure 4-2) was very similar to previously published data

(Kersting et al. 1999, Xia et al. 1999) decreasing monotonically at a much steeper rate in

dark morphs than in light morphs. The distributional pattern of centroid size, size FA and

shape FA appear similar exhibiting right log skew distributions (Figure 4-3). Similar

distributional patterns are observed within temperatures (not shown) as that observed

across temperatures (Figure 4-3).

30

Dark Morph Light Morph
2 5 ......................... . . .
25 -


20 -

DEVELOPMENT
TIME (days) 15 -



1 5 .. .... .... ... 1 ......... 1 ........ ............................................................

10
8 10 12 14 16 18 20 22 24 26

TEMPERATURE (C)

Figure 4-2. Cotton aphid mean development time +1 SE in days in relation to
temperature (n= 531).


































FREQUENCY


100


ND S364 00

CENTROID SIZE


St Dev = 596

T O N = 1822 00

CENTROID SIZE FLUCTUATING ASYMMETRY


J Std Dev= 02
Mean= 056
0 N = 121900
,% *% ,% ,% ,% ,% *o, *,% *% ,% *,%. *% *%

SHAPE FLUCTUATING ASYMMETRY


Figure 4-3. Distribution of isogenic size, size based and shape based FA in monoclonal
cotton aphids grown in controlled environment at different temperatures.
Distributions within each temperature treatment are similar to overall
distributions shown here.











Coefficient of variation for FA was slightly higher for dark morphs (12.5 C =


92.59%, 15 C = 93.02%, 17 C = 88.95%, 19 C = 79.02%, 22.5 C = 80.00% and 25 C =


85.35%). Mean isogenic FA (both size and shape) was highly significantly different


across temperatures (ANOVA F = 6.691, df between group = 4, df within group = 1673,


p < 0.001) in the Gainesville FL clone (Figure 4-4) but not in the Lake Alfred clone


(ANOVA F = 1.992, df between group = 5, df within group = 672, p < 0.078). This is an


indication ofundersampling in the Lake Alfred clone. In the Gainesville clone, mean


centroid size FA (Figure 4-4A) and development time (Figure 4-2) follow a nearly


identical pattern, decreasing rapidly at first then slowing with increased temperature.


1 1 1 1 1 1
10.5 Dark Morph Light Morph
10 A
9.5 -
9
8.5

7.5

6.5

mean isogenic 610 12 14 16 18 20 22 24 26
fluctuating asymmetry
0.068 -
Dark Morph Light Morph
0.066 -
0.064 -
0.062
0.06
0.058
0.056
0.054
0.052 .
10 12 14 16 18 20 22 24 26


TEMPERATURE (C)


Figure 4-4. Mean isogenic FA for (A.) centroid size- based and (B.) Procustes shape-
based) in monoclonal cotton aphids (collected in Gainesville FL) grown on
isogenic cotton seedlings at different temperatures.






71


Mean shape FA was also significantly different across temperature classes

(ANOVA F = 4.863, df within group = 4, df between group = 1673, p = 0.001) but this

difference is due solely to elevated FA in the 12.50C group (Figure 4-4B). Less than one

percent of the variation in FA was due to variation in body size (r = -0.101 for shape FA;

r = 0.088 for size FA). Kurtosis in the shape of the distribution of size based FA (Figure

4-5) was significantly higher in dark morphs than in light morphs (t = -2.21 p = 0.027).

Within each morph (light or dark), kurtosis in the distribution of FA appears to increase

with temperature slightly (Figure 4-5). Measurement error for shape FA was estimated

at2.6% (Lake Alfred clone) and 2.2% (Gainesville clone). For size FA these estimates

were 6.1% (Lake Alfred) and 5.7% (Gainesville).


KURTOSIS


-1


10
10


12 14 16 18 20

TEMPERATURE (C)


Figure 4-5. Kurtosis of size-based FA in monoclonal cotton aphids grown on isogenic
cotton seedlings at different temperatures.


Dark Morph Light Morph ...................................
. ..........


22 24









Discussion


It appears that the prediction of the proximate effect model holds in the case of this

population of cotton aphid, which, in general, is not energetically limited and in this

study, is not genetically variable. Temperature and growth rate (which are positively

correlated in insects) are negatively associated with purely environmentally derived FA

(i.e., developmental noise). This confirms the predictions of several proximate models of

the basis of FA. Furthermore, it appears that centroid size-based FA is a simple function

of development time. Because individual genetic differences in the capacity to buffer

against developmental noise are, in a sense, controlled for in this study by the use of

natural clones, the response of FA to temperature in this study represents a purely

environmental response of FA. The fact that aphids excrete large amount of water and

sugar in the form of honeydew, as well as the lack of a strong correlation between FA and

size also suggests that there is no real energetic "cost" to being large in the aphids in this

study. An important next step will be to compare this result to the association of growth

rate to FA in genetically diverse and energetically limited sexually selected traits where

the predicted association between growth rate and FA would be the opposite of this

study.

It also appears that because the environmental component of size-based isogenic

FA is largely a function of developmental time and temperature, dark morphs of cotton

aphids, which have much longer development time than light morphs, also have

significantly higher FA. The temperature trend in mean FA within both light and dark

morphs, where development times are similar, is not consistent although it appears that

the kurtosis of FA increases slightly with temperature within each morph.









It is very interesting that there is a strong difference in kurtosis between the two

temperature morphs in this study. Previous work suggests that kurtosis is related to

genetic variability in a population (Babbitt in press). The difference observed here in

light of genetic homogeneity in the monoclonal aphid cultures, suggests that there may be

a difference in developmental stability of light and dark aphid morphotypes that is due

primarily to the differential expression of genes in each phenotype.

It is surprising that temperature trends in mean developmental noise are slightly

different regarding whether a size or shape-based approach was used. In populations

where individuals are genetically diverse, both size and shape-based measures of FA are

often correlated to some degree (as they are here too). However, size and shape are

regulated somewhat differently in that cell proliferation is mediated both extrinsically via

cyclin E acting at the G1/S checkpoint of the cell cycle, predominantly affecting size, and

intrinsically at via cdc25/string at the G2/M checkpoint, predominantly affecting pattern

or shape (Day and Lawrence 2000). Extrinsic mechanisms regulate size through the

insulin pathway and its associated hormones (Nijhout 2003) providing a link between

size and the nutritional environment. This may explain why size FA follows

development time more closely than shape FA in this study.

Size is usually less canalized than is shape and therefore more variable. While this

makes the size-based measure of FA generally attractive, it has been found in previous

work (Babbitt et al. 2006) that population averages of size-based FA are often so variable

that they are frequently under-sampled because of the broad and often long tailed

distribution of FA. Shape-based FA does not suffer as much from this problem and so its

estimation is much better although it may be less indicative of environmental influences









and perhaps therefore better suited for genetic studies of FA. Because size FA has been

adequately sampled in this study and because size is more heavily influenced by

environmental factors like temperature, I find that size-based FA is the more interesting

measure of developmental instability in this study.

In general, it is clear that developmental noise is not constant in genetically

identical individuals cultured under near similar environments. The overall distribution

pattern in both size and shape FA is log skewed even within temperature classes. This

suggests that sampling error due solely to random noise can play a very significant role in

FA studies. Furthermore, it appears that the response of FA to the environment is

potentially quite strong. This suggests that FA may indeed be a responsive bioindicator,

however its response will not be very generalizable in environments with fluctuation in

temperature.

In conclusion, the environmental response of developmental noise to temperature

in absence of genetic variability and nutrient limitation supports an important prediction

of theoretical explanations of the proximate basis of FA. This prediction is that FA

should decrease with growth rate (and temperature) in ectothermic organisms. Because it

is only in sexually selected traits that the opposite prediction should hold, that FA should

increase and not decrease with growth rate, the results presented here may offer a useful

method for discriminating between FA that is under sexual selection and FA that is not.














CHAPTER 5
CONCLUDING REMARKS AND RECOMMENDATIONS

In this dissertation, I confirm some existing hypotheses concerning FA and provide

a new explanatory framework that explains alteration in the distribution of fluctuating

asymmetry (FA) and its subsequent effect upon mean FA. The basis of FA and the

influences that shape its response to genes and the environment in populations can be

suitably modeled by stochastic proportional growth in expanding populations of cells on

both sides of the body that are terminated with a small degree of genetically-based

random error. I model stochastic growth with geometric Brownian motion, a random

walk on a log scale. And I also model error in terminating growth with a normal

distribution. The resulting distribution of FA is a lognormal distribution characterized by

power-law scaled tails. Because under a power-law distribution, increased sample size

increases the chance of sampling rare events under these long tails, convergence to the

mean is potentially slowed or even stopped depending on the scaling exponents of the

power-law describing the tails. I demonstrated that the effect of reduced convergence to

the mean is substantial and has probably caused much of the previous work on FA to be

under-sampled (Chapter 1). I have also demonstrated that the scaling effect in the

distribution of FA is directly related to genetic variation in the population. Therefore a

low scaling exponent and also high kurtosis is associated with a large degree of genetic

variation between individuals in their ability to precisely terminate the growth process

(Chapter 2). I also demonstrate that when genetic differences do not exist (where FA is

comprised of only developmental noise) and when development is not limited









energetically, such as in a parthenogenetic aphid clone, FA depends upon developmental

time (Chapter 3).

How Many Samples Are Enough?

The answer to this clearly depends upon whether researchers chose to study FA in

size or FA in shape. The required sample size depends upon the distributional type and

parameters of the distribution of FA. As previously discussed, FA distributions differ

depending upon whether FA is based on size or shape differences between sides. As

demonstrated in chapter 1, multivariate measures of FA based upon shape, such as

Procrustes Distances, tend to have distributional parameters that allow convergence to the

mean that is about five times more rapid than either univariate or multivariate measures

of FA based upon size (Euclidian distance or centroid size). While this information

might seem to favor a shape-based approach, there are some other very important

considerations given below. In the end, sample size must be independently evaluated in

each study depending on the magnitude of asymmetry that one wishes to detect between

treatments or populations. As a general rule, if shape FA is used, 100-200 samples may

suffice, but if size FA is used then many hundreds or even a thousand samples may be

required to detect a similar magnitude difference.

What Measure of FA is best?

The handicap of sampling requirements of size-based FA aside, it appears that

measures of the distributional shape of FA, like kurtosis, scaling exponent in the upper

tail and skewness are strongly related to similar but smaller changes in mean size FA

(chapter 2). Because size FA is most commonly used in past studies of FA, these

measures of distribution shape might allow clearer conclusions to be drawn from

literature reviews and meta-analyses as well as individual studies of FA where the









collection of more data is not possible. My work linking shape of the FA distribution to

the genetic variability of the population will require that future studies of FA which are

focused on the environmental component of FA be controlled for genetic differences

between populations. Where FA is sought as a potential bioindicator of environmental

stress, the potentially differing genetic structure of populations will need to be considered

in order to make meaningful conclusions about levels of FA. Additionally, because in the

absence of genetic variation in monoclonal aphids, size FA is directly related to the total

developmental times of individuals (chapter 3), it may a better choice than shape FA for

studies of environmental influences on FA.

In most organisms, body size is less canalized than body shape. Body size is highly

polygenic and depends upon many environmentally linked character traits. However

body shape or patterning is determined by a sequential progression of the activity of far

fewer genes. Body size is also regulated at a different checkpoint during the cell cycle

than is body pattern formation or shape (Day and Lawrence 2000). Patterning or shape is

regulated at G2/M checkpoint while size is regulated at G1/S checkpoint. The latter

checkpoint is associated with the insulin pathway, linking size to nutrition and hence to

the environment. Because of this the study of environmental responses of FA may

ultimately be best served by using multivariate size-based measures of FA (i.e., centroid

size FA) while at the same time making sure to collect enough samples to accurately

estimate the mean.

In conclusion, multivariate size FA may be the better metric for large studies

interested in the effects of environmental factors upon FA. Where data is harder to come

by (e.g., vertebrate field studies), multivariate shape FA may perform better with the









caveat that shape and pattern are less likely to vary in a population than does size.

Because centroid size must be calculated in order to derive Procrustes distance, both

measures can easily and should be examined together.

Does rapid growth stabilize or destabilize development?

The effects of temperature and growth rate on FA appear to support the proximate

model which predicts that FA declines with increasing growth rates (Figure 4-3). Aphids

from the Gainesville clone decrease mean FA in response to increased temperature and

growth rate. It is likely that the Lake Alfred clone is under-sampled and therefore

estimates of mean FA are not accurate in that sample. The results of the Gainesville

clone, which was sampled adequately, suggests that the prediction of proximate models

of the basis ofFA (Emlen et al. 1993, Graham et al. 1993), that rapid growth should

decrease FA holds true. Moller's hypothesis that rapid growth is stressful because of

incurred energetic costs does not appear to hold in the case of aphids. Of course aphids,

being phloem feeders, generally have access to more water and carbon than they ever

need. This is evidenced by analysis of honeydew composition in this and many other

aphid species. Growth and reproduction in aphids is probably more limited by nitrogen

than by water or energy (carbon). So it would seem that this system may not be a very

good one in which to assess Moller's hypothesis directly. It should be further

investigated whether FA of sexually selected traits in energetically limited, genetically

diverse populations is positively associated with growth rate as is predicted by the

ultimate or evolutionary model presented in Chapter 4.

Can fluctuating asymmetry be a sexually selected trait?

Given that a genetically altered population (chapter 3) demonstrates a consistent

response in the shape of the FA distribution does suggest that a potentially strong genetic









basis for FA exists, but is not easily observable through levels of mean FA. This implies

some heritability exists and possibly opens the door for selection to act upon FA in the

context of mate choice. However, in the future it must be demonstrated, with appropriate

sample sizes, that the level of FA in the laboratory can be altered by selection for

increased or decreased FA. The opposing predictions of the proximate or mechanistic

and ultimate (evolution through sexual selection) explanations of FA, regarding the

relationship of temperature, growth rate and FA may offer a method for detecting sexual

selection upon FA (chapter 4). If FA has evolved as an indicator of good genes, and

hence has related energetic costs, the expectation that FA should increase with growth

rate and temperature (in ectotherms) is plausible. Otherwise, ifFA is caused by random

accumulation of error during development, then the expectation is reversed. FA should

decrease with growth rate and temperature as I have demonstrated in a clonal population

of aphids that are not energetically limited (chapter 4).

Is fluctuating asymmetry a valuable environmental bioindicator?

Given that the genetic structure of a population (chapter 3) has potentially strong

effects upon the shape and location (mean) of the distribution of FA, the usefulness of FA

as a bioindicator of environmental stress has to be questioned. Average levels of FA of

populations would be difficult to interpret or compare without additional information

regarding genetic heterogeneity. However, provided that the study of FA and population

genetics were undertaken simultaneously, this problem could be avoided. So FA could

still be of use in this regard, however its expense and ease of use compared to other

bioindicators would need to be reassessed.









Scaling Effects in Statistical Distributions: The Bigger Picture

This dissertation demonstrates that underlying distributions of some biological data

can contain partial self-similarity or power-law scaling. The proper model for describing

data such as this sits at a midpoint between those models used in classical statistics and

those of statistical physics: Levy statistics (Bardou et al. 2003). To my knowledge, this

work represents the first application of such a model in biology.

The sources of power-law scaling in the natural sciences are diverse. Sornette

(2003) outlines 14 different ways that power-laws can be created, some of which are very

simple. The hypothesis that all power-law scaling in nature is due to a single phenomena

such as self-organized criticality (SOC) (Bak 1996, Gisiger 2000) or highly optimized

tolerance (HOT) (Newman 2000) is unlikely. Reed (2001) suggests that the model I have

used here, stochastic proportional growth that is observed randomly, may explain a great

deal of power-law scaled size distributions formerly speculated to have a single cause

like SOC or HOT. These phenomena include distributions of city size (Zipf's law),

personal income (Pareto's Law), sand particle size, species per genus in flowering plants,

frequencies of words in sequences of text, sizes of areas burnt in forest fires, and species

body sizes, just to name a few examples.

One of the most common ways that power-laws can be obtained is by combining

exponential functions. This is effect is also observed when positive exponential/

geometric/proportional growth is observed with a likelihood described by to a negative

exponential function such as in Reed and Jorgensen's (2004) model for generating size

distributions. Power laws generated by combing exponential functions are also present in

the statistical mechanics of highly interactive systems (e.g., SOC) where the correlation

in behavior between two nodes or objects in an interaction network or lattice decreases









exponentially with the distance between them while the number of potential interaction

paths increases exponentially with the distance between them (Stanley 1995). In

Laplacian fractals, or diffusion limited aggregations observed in chemical

electrodeposition and bacterial colony growth (Viscek 2001) there may be a similar

interplay between exponential growth (doubling) and the diffusion of nutrients supporting


growth which are governed by the normal distribution, of the form y = e

The exponential function holds a special place in the natural sciences. Any

frequency dependent rate of change in nature, or in other words, any rate of change of

something that is dependent on the proportion of that something present at that time, is

described by the exponential function. Therefore it finds application to many natural

phenomena including behavior of populations, chemical reactions, radioactive decay, and

diffusion just to name a few. It seems only fitting that many of the power-laws we

observe in the natural sciences probably owe their existence to the interplay of

exponential functions, one of the most common mathematical relationships observed in

nature.

As far as we can ascertain from the recording of ancient civilizations, human beings

have been using numbers for at least 5000 years if not longer. And yet the concepts of

probability are a relatively recent human invention. The idea first appears in 1545 in the

writings of Girolamo Cardano and is later adopted by the mathematicians, Galileo,

Fermat, Pascal, Huygens, Bernoulli and de Moivre in discussions of gambling over the

next several hundred years. The concepts of odds and of random chance are not

generalized until the early twentieth century by Andreyevich Markov and not formalized

into mathematics until the work of Andrei Kolmogorov in 1946.









The role of uncertainty in nature is yet to be resolved. Does uncertainty lie only

with us or does it underlie the very fabric of the cosmos? Most 19th century scientists

(excepting perhaps Darwin) believed that the universe was governed by deterministic

laws and that uncertainty is solely due to human error. The first application of statistical

distributions by Carl Frederick Gauss and Pierre Simon Laplace were concerned only

with the problem of accounting for measurement error in astronomical calculations. The

more recent view, that uncertainty is something real, was largely the work of early

twentieth century quantum theorists Werner Heisenberg and Erwin Schrodinger and the

statistical theory embodied in the work of Sir Ronald Fisher. The basic conceptual

revolution in modem physics was that observations cannot be made at an atomic level

without some disturbance, therefore while one might observe something exactly in time,

one cannot predict how the act of observing will affect the observed in the future with

any degree of certainty (at least at very small scales). This is Heisenberg's Uncertainty or

Indeterminancy Principle.

Scientists observe natural "laws" only through emergent properties of many atoms

observed at vastly larger scales where individual behavior is averaged into a collective

whole. Jakob Bernoulli's Law of Large Numbers, Abraham de Moivre's bell shaped

curve, and Sir Ronald Fisher's estimation of the mean are also examples of how scientists

rely upon emergent properties of large systems of randomly behaving things in order to

make sense of what is thought to be fundamentally uncertain world. However, this

probabilistic view of the natural world has been challenged in recent decades by some

mathematicians who are again championing a deterministic view of nature. Observations

by Edward Lorenz, Benoit Mandelbrot and others, of simple and purely deterministic









equations that behave in complex unpredictable ways, has led to a revival in the

deterministic view among mathematicians; in this case, uncertainty is a product of human

inability to perceive systems that are extremely sensitive to initial starting conditions.

For this reason, this new view is often called deterministicc chaos". And so the question

as to whether uncertainty is real or imagined is yet to be resolved by modern science.

One remarkable observation of this latest revolution in mathematics is the frequent

occurrence of scale invariance or power-law scaling in systems that exhibit this sort of

complex and unpredictable behavior. And so just as the normal distribution or bell curve

is an emergent property governing the random behavior of independent objects, the

power-law appears to be an emergent property of nature as well; one that seems not only

to often to appear in the behavior of interacting objects (i.e., critical systems) but in

systems where growth in randomly observed as well. Statistical physics now recognizes

two classes of "stable laws", one that leads to the Gaussian or normal distribution and

another that is Levy or power-law distributed. In the former class, emergent behavior is

governed by commonly occurring random events while in the latter class the behavior of

the group is governed by a few rarely occurring random events. As I have already

reviewed earlier, we find that convergence to the mean under these two frameworks can

be radically different. Yet both are present in the natural world, and as I have shown in

my work, both of these classes of behavior can underlie naturally occurring distributions,

causing partial scale invariance in real data. It is my conviction that the biological

sciences must in the future adopt the statistical methods of working under both of these

frameworks and not simply make assumptions of normality whenever and wherever

random events are found to occur.














APPENDIX A
LANDMARK WING VEIN INTERSECTIONS CHOSEN FOR ANALYSISOF
FLUCTUATING ASYMMETRY


Figure Al. Six landmark locations digitized for Aphis gossipyii


Figure A2. Six landmark locations digitized for Apis mellifera






85















Figure A3. Six landmark locations digitized for Chrysosoma crinitus


Figure A4. Eight landmark locations digitized for Drosophila simulans













APPENDIX B
USEFUL MATHEMATICAL FUNCTIONS

The following is a list of the probability density functions for candidate models of
the distribution of fluctuating asymmetry. Log likelihood forms of these functions were
maximized to obtain best fitting parameters of each model for our data.
Asymmetric Laplace Distribution

(see Kotz et al. 2001)

f(x) = ( 1)e eo for x > 0

f(x) = ((1 )) for x < 0

where 0 = location, c = scale and K = skew index (Laplace when K = 1)
Half-Normal Distribution

f(x) = ( )( )e'

where [= minimum data value and 0 = dispersion
Lognormal Distribution

(see Evans 2000 or Limpert et al. 2001)

f(x) = 1 e( (logx v) 2-2)
xr(2r)1/2
where v = location and T = shape or multiplicative standard deviation
Double Pareto Lognormal Distribution

(see Reed and Jorgensen 2004)

f(x) = -- g(log(x))
given the normal-Laplace distribution

g(y) = -(At) [R(a -Y)+ R(P. ) + )]
where the Mill s ratio R(z) is









1- O(z)
R(z) =

and where D is the cumulative density function and q is probability density function for

standard normal distribution N(0,1), where a and 0 are parameters that control power-law

scaling in the tails of the lognormal distribution.

The limiting forms of the double Pareto lognormal are the left Pareto lognormal

(a = o ), right Pareto lognormal distributions (f = oo ), and lognormal distributions

(a = o, f = o) with Pareto tails on only the left side, only the right side, or on neither

side, respectively.

A description of Reed and Jorgensen's generative model of double Pareto lognormal size
distribution.

Reed and Jorgenson's (2004) generative model begins with the Ito stochastic

differential equation representing a geometric Brownian motion given below.

dX = uXdt + crXdw

with initial state X(0) = XO distributed lognormally, log XO N(v,'2). After Ttime units

the state X(T) is also distributed lognormally with log X(T) N(v + (La-C2/2)T,T2 + C2T).

The time T, at which the process is observed, is distributed with densityfT(t) = Xe-Xt

where X is a constant rate. The double Pareto lognormal distribution is generated when

geometric Brownian motion is sampled repeatedly at time t with a negative exponential

probability.




Full Text

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CHARACTERIZATION OF THE DISTRIBUTION OF DEVELOPMENTAL INSTABILITY By GREGORY ALAN BABBITT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Gregory Alan Babbitt

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This document is dedicated to my grandmothe r, Sarah Miller, who taught me to admire the natural world.

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iv ACKNOWLEDGMENTS I would like to thank Rebecca Kimball, Susan Halbert, Bernie Hauser, Jane Brockmann, and Christian Klingenberg for helpful discussions and comments; Susan Halbert and Gary Steck (Division of Plant Industry, State of Florida) for specimen ID; Marta Wayne for use of her microscope and digital camera; Glenn Hall (Bee Lab, University of Florida) for a plentiful suppl y of bees; and the Department of Entomology and Nematology (University of Florida) for use of its environmental chambers. I thank Callie Babbitt for help in manuscript prepar ation and much additional love and support.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT......................................................................................................................x ii CHAPTER 1 SEARCHING FOR A CONSISTENT INTERPRETATION OF DEVELOPMENTAL INSTABILITY? A GENERAL INTRODUCTION................1 2 ARE FLUCTUATING ASYMMETRY STUDIES ADEQUATELY SAMPLED? IMPLICATIONS OF A NEW MO DEL FOR SIZE DISTRIBUTION......................10 Introduction.................................................................................................................10 Developmental Stability: Definition, Measurement, and Current Debate..........10 The Distribution of Fluctuating Asymmetry.......................................................12 Methods......................................................................................................................17 Results........................................................................................................................ .22 Discussion...................................................................................................................30 The Distribution of FA........................................................................................30 Sample Size and the Estimation of Mean FA......................................................32 The Basis of Fluctuating Asymmetry..................................................................33 Conclusion...........................................................................................................34 3 INBREEDING REDUCES POWER-LAW SCALING IN THE DISTRIBUTION OF FLUCTUATING ASYMMETRY: AN EXPLANATION OF THE BASIS OF DEVELOPMENTAL INSTABILITY........................................................................36 Introduction.................................................................................................................36 What is the Basis of FA?.....................................................................................36 Exponential Growth and Non-No rmal Distribution of FA..................................38 Testing a Model for the Basis of FA...................................................................40 Methods......................................................................................................................41 Model Development............................................................................................41 Simulation of geometric Brownian motion..................................................41

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vi Simulation of fluctuating asymmetry...........................................................44 Inbreeding Experiment........................................................................................45 Morphometric analyses.......................................................................................48 Model selection and inference.............................................................................49 Results........................................................................................................................ .49 Model Simulation................................................................................................49 Experimental Results...........................................................................................50 Model Selection and Inference............................................................................52 Discussion...................................................................................................................54 Revealing the Genetic Component of FA...........................................................54 Limitations of the Model.....................................................................................56 The Sources of Scaling........................................................................................56 Potential Application to Cancer Screening..........................................................57 Conclusion...........................................................................................................58 4 TEMPERATURE RESPONSE OF FLUCTUATING ASYMMETRY TO IN AN APHID CLONE: A PROPOSAL FOR DETECTING SEXUAL SELECTION ON DEVELOPMENTAL INSTABILITY........................................................................60 Introduction.................................................................................................................60 The Genetic Basis of FA.....................................................................................60 The Environmental Basis of FA..........................................................................61 Temperature and FA in and Aphid Clone...........................................................63 Methods......................................................................................................................65 Results........................................................................................................................ .68 Discussion...................................................................................................................72 5 CONCLUDING REMARKS AND RECOMMENDATIONS..................................75 How Many Samples Are Enough?.............................................................................76 What Measure of FA is best?......................................................................................76 Does rapid growth stabilize or destabilize development?..........................................78 Can fluctuating asymmetry be a sexually selected trait?............................................78 Is fluctuating asymmetry a valuab le environmental bioindicator?.............................79 Scaling Effects in Statistical Di stributions: The Bigger Picture.................................80 APPENDIX A LANDMARK WING VEIN INTERSECTIONS CHOSEN FOR ANALYSISOF FLUCTUATING ASYMMETRY..............................................................................84 B USEFUL MATHEMATICAL FUNCTIONS.............................................................86 Asymmetric Laplace Distribution...............................................................................86 Half-Normal Distribution...........................................................................................86 Lognormal Distribution..............................................................................................86 Double Pareto Lognormal Distribution......................................................................86

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vii LIST OF REFERENCES...................................................................................................88 BIOGRAPHICAL SKETCH.............................................................................................96

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viii LIST OF TABLES Table page 2-1. Maximized log-likelihood (MLL), numbe r of model parameters (P) and Akaike Information Criterion differences ( AIC) for all distribut ional models tested. Winning models have AIC difference of zer o. Models with nearly equivalent goodness-of-fit to winners are underscored ( AIC <3.0). 4.0 < AIC <7.0 indicates some support for specified model. AIC >10.0 indicates no support (Burnham and Anderson 1998). Distances between landmarks (LM) used for first univariate size FA, aphids LM 1-2, bees LM 1-4 and l ong-legged fly LM 36 and for second univariate FA aphids LM 2-3, bees LM 2-6 and long-legged fly LM 4-5......................................................................................................................23 2-2. Best fit parameters for models in Table 2-1. Parameters for univariate size FA are very similar to multivariate size FA and are not shown. Skew indexes for asymmetric Laplace are also not shown...................................................................24 3-1. Distribution parameters and model fit for multivariate FA in two wild populations and four inbred lines of Drosophila simulans and one isogenic line of Drosophila melanogaster Model fits are AIC for unsigned centroid size FA (zero is best fit, lowest number is next best fit).................................................51

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ix LIST OF FIGURES Figure page 2-1. Schematic representation of mathem atical relationships between candidate models for the distribution of fluctu ating asymmetry. Mixtures here are continuous................................................................................................................14 2-2. Distribution of multivariate shape FA of A) cotton aphid ( Aphis gossipyii ) B) domestic honeybee ( Apis mellifera ) and C) long-legged fly ( Chrysosoma crinitus ). Best fitting lognormal (dashed line, lower inset), and double Pareto lognormal (solid line, uppe r inset) are indicated......................................................25 2-3. Distribution of multivariate centroid size FA of A) cotton aphid ( Aphis gossipyii ) B) domestic honeybee ( Apis mellifera ) and C) long-legged fly ( Chrysosoma crinitus ). Best fitting half-normal (dashed line, lower inset) and double Pareto lognormal distribution (solid line upper inset) are indicated..................................26 2-4. Distribution of univariate uns igned size FA of A) cotton aphid ( Aphis gossipyii ) B) domestic honeybee ( Apis mellifera ) and C) long-legged fly ( Chrysosoma crinitus ). Best fitting half-normal (dashe d line, lower inset) and double Pareto lognormal distribution (solid line upper inset) are indicated..................................27 2-5. Distribution of sample sizes (n) from 229 fluctuating asymme try studies reported in three recent meta-analyses (Vollest ad et al. 1999, Thornhill and Mller 1998 and Polak et al. 2003). Only five studies had sample sizes greater than 500 (not shown)......................................................................................................................28 2-6. Relationship between sample size and % error for estimates of mean FA drawn from best fitting size (dashed line) and shape (solid line) distributions using 1000 draws per sample size. All ru ns use typical winning double Pareto lognormal parameters (shape FA = -3.7, = 0.2, = 1000, = 9; for size FA = 1.2, = 0.7, = 4.0, = 4.0).................................................................................29 2-7. The proportion and percen tage (inset) of individuals with visible developmental errors on wings (phenodeviants) are show n for cotton aphids (A) and honeybees (B) in relation to distribution of shape FA (Procrustes distance). Average FA for both normal and phenodeviant aphids (C) and honeybees (D) are also given.........30 3-1. Ordinary Brownian motion (low er panel) in N simulated by summing independent uniform random variables (W) (upper panel)......................................42

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x 3-2. Geometric Brownian motion in N and log N simulated by multiplying independent uniform random variables. This was generated using Equation 1.5 with C = 0.54............................................................................................................43 3-3. Geometric Brownian motion in N a nd log N with upward drift. This was generated using Equation 1.5 with C = 0.60............................................................44 3-4. Model representation of Reed and Jo rgensens (2004) physic al size distribution model. Variable negativ e exponentially distributed stopping Times of random proportional growth (GBM with C = 0.5) create double Pareto lognormal distribution of size....................................................................................................46 3-5. A model representation of developm ental instability. Normally variable stopping times of random proportional gr owth (GBM with C > 0.5) create double Pareto lognormal di stribution of size...........................................................47 3-6. Simulated distributions of cell popul ation size and FA for different amounts of variation in the termination of growth (variance in normally distributed growth stop time). The fit of simulated data to the normal distribution can determined by how closely the plotted points foll ow the horizontal line (a good fit is horizontal)................................................................................................................50 3-7. Distribution of fluctuating asymme try and detrended fit to normal for two samples of wild population collected in Ga inesville, FL in summers of 2004 and 2005 and four inbred lines of Drosophila simulans derived from eight generations of full-sib crossing of the w ild population of 2004. Also included is one isogenic line of Drosophila melanogaster (mel75). All n = 1000. The fit of data to the normal distribution can dete rmined by how closely the plotted points follow the horizontal line (a good fit is horizontal).................................................53 4-1. Predicted proximate and ultimate leve l correlations of te mperature and growth rate to fluctuating asymmetry are different Ultimate level (evolutionary) effects assume energetic limitation of individua ls in the system. Proximate level (growth mechanical) effects do not. No tice that temperature and fluctuating asymmetry are negatively correlated in the proximate model while in the ultimate model they ar e positively correlated..........................................................64 4-2. Cotton aphid mean development time SE in days in relation to temperature (n= 531)....................................................................................................................68 4-3. Distribution of isogenic size, size based and shape based FA in monoclonal cotton aphids grown in controlled en vironment at different temperatures Distributions within each temperature treatment are similar to overall distributions shown here...........................................................................................69 4-4. Mean isogenic FA for (A.) centroid si zebased and (B.) Pr ocustes shape-based) in monoclonal cotton aphids (collected in Gainesville FL) grown on isogenic cotton seedlings at different temperatures................................................................70

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xi 4-5. Kurtosis of size-based FA in monoc lonal cotton aphids gr own on isogenic cotton seedlings at differe nt temperatures...........................................................................71

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xii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CHARACTERIZATION OF THE DI STRIBUTION OF DEVELOPMENTAL INSTABILITY By Gregory Alan Babbitt May 2006 Chair: Rebecca Kimball Cochair: Benjamin Bolker Major Department: Zoology Previous work on fluctuating asymmetr y (FA), a measure of developmental instability, has highlighted its controversial relationship w ith environmental stress and genetic architecture. I suggest that conflict may derive from th e fact that the basis of FA is poorly understood and, as a consequen ce, the methodology for FA studies may be flawed. While size-based measures of FA have been assumed to have half-normal distributions within populations, developmen tal modeling studies have suggested other plausible distributions for FA. Support for a non-normal distribution of FA is further supported by empirical studies that often record leptokurtic (i.e., fat or long-tailed) distributions of FA as well. In this disse rtation, I investigate a series of questions regarding the both the basis and distributi on of FA in populations. Is FA normally distributed and therefore likel y to be properly sampled in FA studies? If not normal, what candidate model distribution best fits th e distribution of FA? Is the shape of the distribution of FA similar to a simple and specific growth model (geometric Brownian

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xiii motion)? Does reducing individual variati on in populations through inbreeding affect follow the prediction of this model? Ho w does this shape respond to environmental factors such as temperature when ge netic variation is controlled? In three species of insects (cotton aphid, Aphis gossipyii Glover; honeybee, Apis mellifera ; and long-legged fly, Chrysosoma crinitus (Dolichipodidae)), I find that FA was best described by a double Pareto lognor mal distribution (DPLN), a lognormal distribution with power-law tails. The larg e variance in FA under this distri bution and the scaling in the tails both act to slow convergence to the mean, suggesting that many past FA studies are under-sampled when the dist ribution of FA is assumed to be normal. Because DPLN can be generated by geometric Brownian motion, it is ideal for describing behavior of cell populations in growing tissue. I demonstrated through both a mathematical growth model and an inbreeding experiment in Drosophila simulans that the shape of the distribution of FA is highl y dependent on the level of genetic redundancy or heterogeneity in a populati on. In monoclonal lines of cott on aphids, I also demonstrate that FA decreases with temp erature and that a shift in kurtosis is associated with temperature induced phenotypic plasticity. Th is supports the prediction of a proximate model for the basis of FA and also suggests shape of the distribution of FA responds to environmentally induced changes in gene expression on the same genetic background.

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1 CHAPTER 1 SEARCHING FOR A CONSISTENT INTERPRETATION OF DEVELOPMENTAL INSTABILITY? A GENERAL INTRODUCTION Everywhere, nature works true to sc ale and everything ha s its proper size. -DArcy Thompson There have been many incarnations of th e idea that stability and symmetry are somehow related. Hippocrates (460-377 B.C. ) was the first to postulate internal corrective propertie s that work in the presence of disease. Waddington (1942) suggested existence of similar homeostatic buffering agai nst random and presumably additive errors occurring during development. The term f luctuating asymmetry, first introduced by Ludwig (1932), was later adopted by Mather (1953), Reeve (1960) and Van Valen (1962) to describe a measurable form of morphological noise repres enting a hypothetical lack of buffering that is always present during develo pment of organisms. Recently, biologists have become very interested both in fluctu ating asymmetrys poten tial usefulness as a universal bioindicator of environmental health (Parsons 1992) and in its potential as an indicator or even an overt signal of an indi viduals overall genetic quality (Mller 1990). However, over a decade of work has left the field with no clear relationship between increased fluctuating asymmetry and either e nvironmental or genetic stress (Bjorksten et al. 2000, Lens et al. 2002). Despite this fact fluctuating asymmetry is still often assumed to indicate of developmental instability. Recently, Debat and David (2001) page 560 define developmental stability as a set of mechanisms historically selected to keep the phenotype constant in spite of small random developmental i rregularities potentially

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2 inducing slight differences among homologous parts wi thin individuals. ( I have italicized the aspects of this definition that I think are left undefined. ) Fluctuating asymmetry is defined and meas ured as the average right minus left difference in size or shape of morphological characters in a population and has been generally accepted as an indicator of devel opmental instability because both sides of a bilaterally symmetric organism have been de veloped by the same genetic program in the same environment (Mller and Swaddle 1997). Fluctuating asymmetry is measured as left-right side differences in the size or shape of paired b ilaterally symmetric biological structures of organisms. In a character trait that demonstrates fluctu ating asymmetry, it is assumed that the distribution of signed leftright differences is near ze ro and that there is no selection for asymmetry (Palmer and Str obek 1986, 2003). Other types of asymmetry do exist and are thought to indi cate selection against symmet ry. Directional asymmetry denotes a bias towards left or right side dness that causes the population mean to move away from zero. In antisymmetr y, left or right side biases occur equally at the individual level creating a population that is bimodal or platykurtic. While much attention has been directed toward the possibl e genetic basis of fluctuating asymmetry (reviewed by Leamy and Klingenberg 2005, Woolf and Markow 2003), response of fluctuating asymmetry to stress (reviewed by Hoffman and Woods 2003) and correlation of fluctuating asymmetr y with mate choice (Mller and Swaddle 1997); little scientific effort ha s been directed toward investig ating its basis or origin at levels of organization lower than the individu al. A few theoretical explanations for the basis of fluctuating asymmetry have been developed (reviewed by Klingenberg 2003). Only two of these offer a causal explanati on for the increased levels of fluctuating

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3 asymmetry that are sometimes observed under periods of environmental stress. While not mutually exclusive, these two explanations differ at the organizational level at which they are thought to act. Mller and Pomiankowski (1993) fi rst suggested that strong natural or sexual selection can remove regulatory steps controlling the symmetric development of certain traits (e.g., morphology us ed in sexual display). They suggest that with respect to sexually selected traits (a nd assuming that they are somehow costly to produce), individuals may vary in their ability to buffer ag ainst environmental stress in proportion to the size of their own energetic rese rves. These reserves are often indicative of the genetic quality of indi viduals. Therefore, high geneti c quality is expected to be associated with low fluctuating asymmetry. Emlen et al. (1993) present another and more proximate explanation of the basis of fluctuating asymmetry. They do not invoke sexual selection but hypothesize that fluctu ating asymmetry is due largely to the nonlinear dynamics of signaling a nd supply that may occur during growth. Here fluctuating asymmetry is thought to result from th e scaling up of compounding temporal asymmetries in signaling between cells duri ng growth. In their model, hypothetical levels of signaling compounds (morphogens) and or growth precursors used in the construction of cells vary randomly over time. When growth suffers less interruption, in other words, when it occurs faster and under le ss stress, there is al so less complexity (and fractal dimension) in the dynamics of si gnaling and supply. This is should reduce fluctuating asymmetry. Only a few other models have since b een proposed. A model by Graham et al. (1993) suggests that fluctuati ng asymmetry in the individual can also be the net result of compounding time lags and chaotic behavior be tween hormonally controlled growth rates

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4 on both sides of an axis of bilateral symmet ry. More recently Klingenberg and Nijhout (1999) present a model of morphogen diffusi on and threshold response that includes genetic control of each component. They de monstrate that fluctuating asymmetry can result from genetically modulated expression of variation that is en tirely non-genetic in origin. In other words, even without specific genes for fluctuating asymmetry, interaction between genetic and non-genetic sources of variation (G x E) can cause fluctuating asymmetry (Klingenberg 2003). While all these theories for the basis of fluctuating asymmetry have proven useful in making some predictions about fl uctuating asymmetry in relation to sexual selection and growth rate/trait size, none are grounded in any known molecular mechanisms. The search for any single molecular mechanism that stabilizes the developmental process has proven elusive. A recent candidate was the heat shock protein, Hsp90, which normally target conformati onally plastic proteins that act as signal tranducers (i.e., molecular sw itches) in many developmental pathways (Rutherford and Lundquist 1998). Because Hsp90 recognizes protei n folding, it can also be diverted to misfolded proteins that are denatured during environmental stress. Therefore, Hsp90 can potentially link the developmental process to the environment and these authors suggest it may also capacitate the evolution of novel morphology during times of stress by revealing genetic variation previously hidden to selection in non-stre ssful environments. However, additional research by Milton et al. (2003) shows that while Hsp90 does buffer against a wide range of morphologic changes and does mask the effect of much hidden genetic variation in Drosophila it does not appear to affect average levels of fluctuating asymmetry through any single Hsp90 depe ndent pathway or process.

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5 It is said that wisdom begins with the naming of things. The case of fluctuating asymmetry reminds us that the act of giving na mes to things in science can, in fact, lend a false impression that we have achieved a tr ue understanding of that which has been named. While fluctuating asymmetry has had several specific descriptive definitions, we do not really know how to define it at a f undamental level because we do not understand exactly how development is destabilized under certain conditions of both gene and environment. All we can say for now is that fluctuating asymmetry is a mysterious form of morphological variation. Mary Jane West-Eberhard ( 2003) has dubbed it the dark side of variation because it may represent th at noisy fraction of the physical-biological interface that is still free of selection, and not under direct control of the gene. Any study of natural variation would do well to begin by simply observing its shape or its distribution in full. In nature, statistical distributions come mostly in two flavors: those generated by large systems of independent additive components and those generated by large systems of intera cting multiplicative components (Vicsek 2001, Sornette 2003). When large systems are composed of independent subunits, random processes result in the normal distribution, th e cornerstone of classical statistics. The normal distribution is both unique and extrem e in its rapidly decaying tails, its very strong central tendency and its sufficient de scription by just two parameters, the mean and variance, making it, in this sense, th e most parsimonious description of random variation. However, the normal distribution does not describe all kinds of stochastic or random behavior commonly observed in the na tural sciences. When subunits comprising large systems interact, random processes are best described by models that underlie statistical physics (sometimes called Levy st atistics) (Bardou et al 2003. Sornette 2003).

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6 Fat-tailed distributions are often the result of propagation of error in the presence of strong interaction. These type s of distributions include th e power function, Pareto, Zipf and the double Pareto or log-Laplace distri butions, all characterized by a power law and an independence of scale. Collective effect s in complex interacting systems are also often characterized by these power laws (Wilson 1979, Stanley 1995). Examples include higher order phase transitions, self-organized criticality and percolation. During second order phase transition at the critical temperature between physical phases, external perturbation of the network of microscopic interactions between molecules results in system reorganization at a macroscopic level far above that of interacting molecules (e.g., the change from water to ice). This results in collective imitation that propagates among neighboring molecules over long distances. Exactly at these critical temperatures, imitation between neighbors can be observed at all scales creating regions of similar behavior that occur at all sizes (Wilson 1979). Thus, a self-similar power-law manifests itself in the interacting systems susceptibility to perturbation and re sults, in this case, from the multiplicity of intera ction paths in the interacti on network (Stanley 1995). As the distance between two objects in a netw ork increases, the nu mber of potential interaction pathways increas es exponentially and the corr elation between such paths decreases exponentially. The constant cont inuous degree of change represented by the power law is the result of a combined eff ect between both an exponentially increasing and decreasing rate of change. This highlig hts the fact that power laws can be easily manifested from combinations of exponen tial functions which are very common to patterns of change in many natural popul ations (both living and non-living).

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7 Therefore, it is important to note that power laws in statistica l distributions do not have to always indicate strong interacti on in a system and that many other simple mechanisms can create them (Sornette 2003), especially where the behavior of natural populations are concerned. For example, an apparently common method by which power laws are generated in nature is when st ochastic proportional (geometric) growth is observed randomly in time (Reed 2001). Power law size distributions in particle size, human population size, and economic factors are all potentially explained by this process (Reed and Jorgensen 2004). Here we also have exponential increa se in size opposing an exponential decrease in the pr obability of observation or te rmination that results in a gently decaying power law. This process can also explain why power law scaling can occur through the mixing of certain distribu tions where locally exponentially increasing and decreasing distributions overlap. For example, superimposing lognormal distributions results in a lognormal distri bution with power law tails (Montroll and Shlesinger 1982,1983). Given that exponential relati onships are so common in th e natural world, we should assume that in observing any large natural population outside of an experiment, there is probably some potential for a power-law scaling effect to occur. Therefore some degree of non-normal behavior may be likely to be observed, often in the underlying distributions tail. If we assume an u nderlying normal distribution, and sample it accordingly, we are likely to under-sample th is tail. And so we rarely ever present ourselves with enough data to challenge our assumption of normality, and we risk missing the chance to observe a potentially important aspect of natural variation.

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8 Until now, the basis of fluctuating asymmetry has been addressed only with very abstract models of hypothetical cell signaling, or at the leve l of selection working on the organism with potential mech anism remaining hypothetical. In this dissertation, the underlying common theme is that fluctuating asymmetry must first and foremost be envisioned as a stochastic process occurri ng during tissue growth, or in other words, occurring within an exponen tially expanding populat ion of cells. This expansion process can be modeled by stochastic proportional (geo metric) growth that is terminated or observed randomly over time. As will be explained in subsequent chapters, this generative process can naturally lead to variation that distributes according to a lognormal distribution with power laws in both tails (Reed 2001). In chapter 2, I examine the distribution of fluc tuating asymmetry in the wings of three species of insects (cotton aphid, Aphis gossipyi Glover honeybee, Apis mellifera and long-legged flies, Chrysosoma crinitus ) and test various candida te models that might de scribe the statistical distribution of fluctuating asym metry. I then address whether, given the best candidate model, fluctuating asymmetry studies have b een appropriately sampled. I suggest that much of the current controversy over fluctuat ing asymmetry may be due to the fact that past studies have been under-sample d. In chapter 3, I extend and test a phenomenological model for fluctuating asymme try that is introduced in chapter 2. I present the model and then examine some of its unique predictions concerning the effects of inbreeding on the shape of the dist ribution of fluctuating asymmetry in Drosophila I present evidence that the gene tic structure of a population can have a profound effect on the scaling and shape of the observed distribut ion of fluctuating asym metry. In chapter 4, I characterize the pattern of developmental noise (fluctuatin g asymmetry in the absence

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9 of genetic variation) in two monoclona l populations of cotton aphid cultured under graded environmental temperatures. I inves tigate how developmental noise is altered by this simple change in the environment. I present evidence that the environmental response of size-based FA is di rectly related to developmental time. Lastly, I conclude with a review of my major findings in the context of the introduc tion I have presented here.

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10 CHAPTER 2 ARE FLUCTUATING ASYMMETRY STUDIES ADEQUATELY SAMPLED? IMPLICATIONS OF A NEW MO DEL FOR SIZE DISTRIBUTION Introduction Developmental Stability: Definition, Measurement, and Current Debate Developmental stability is maintained by an unknown set of mechanisms that buffer the phenotype against small random pert urbations during development (Debat and David 2001). Fluctuating asymmetry (FA) the most commonly used assay of developmental instability, is defined either as the average deviation of multiple traits within a single individual (Van Valen 1962) or the deviation of a single trait within a population (Palmer and Strobek 1986, 2003; Parsons 1992) from perfect bilateral symmetry. Ultimately, an individuals developm ental stability is the collective result of random noise, environmental influences, and th e exact genetic archite cture underlying the developmental processes in that indivi dual (Klingenberg 2003; Palmer and Strobek 1986). Extending this to a populat ion, developmental stability is the result of individual variation within each of these three components. Currently, there is conflict in the literature regarding th e effect of both environment and genes on the developmenta l stability of populations. Th e development of bilateral symmetry appears to be destabilized to vari ous degrees by both e nvironmental stressors (review in Mller and Swaddle 1997) and certain genetic arch itectures (usually created by inbreeding: Graham 1992; Lerner 1954; Mather 1953; Messier and Mitton 1996; review by Mitton and Grant 1984). While th e influence of inbreeding on FA is not

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11 consistent (Radwan 2003; Carchi ni et al. 2001; Fowler and Whitlock 1994; Leary et al. 1983,1984; Lens et al. 2002; Perfectti and Camach i 1999; Rao et al. 2002; Vollestad et al. 1999), it has led biologists to us e terminology such as genetic stress or developmental stress when describing inbred populations (Clarke et al. 198 6 and 1993; Koehn and Bayne 1989; Palmer and Strobek 1986). While genetic and environmental stressors have been shown to contribute to developmental instability and FA, the full pict ure is still unclear (B jorksten et al. 2000; Lens et al. 2002). While FA has been proposed as a universal indicator of stress within individual organisms (Parsons 1992), its util ity as a general indicator of environmental stress has been contentious (Bjorksten et al. 2000; Ditchkoff et al. 2001; McCoy and Harris 2003; Merila and Bjorklund 1995; M ller 1990; Rasmuson 2002; Thornhill and Mller 1998; Watson and Thornhill 1994; Wh itlock 1998). Despite many studies, no clear general relationship betw een environmental stress and FA has been demonstrated or replicated through experimentation across different taxa (Bjorksten et al. 2000; Lens et al. 2002). Furthermore, the effects of stress on FA appear to be not only species-specific but also trait-specific and stress-sp ecific (Bjorksten et al. 2000). Several meta-analyses have attempted to unify individual studies on the relation of sexual sele ction, heterozygosity, and trait specificity to FA (Polak et al. 2003; Thornhill and Mller 1998; Vollestad et al. 1999); while some weak general effects have been found, their biolog ical significance is still unresolved. Taken together, the ambiguity of the resu lts from FA studies suggests unresolved problems regarding the definition and/or m easurement of FA. The distribution and overall variability of FA are sometimes disc ussed with regards to repeatability in FA

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12 studies (Whitlock 1998; Palmer and Strobeck 2003), but is seldom a primary target of investigation. Until we can quantify FA mo re reliably and understand its statistical properties, the potential for misinterpret ation of FA is likel y to persist. The Distribution of Fluctuating Asymmetry Although it is always risky to infer unde rlying processes from observed patterns, careful examination of the distribution of FA in large samples may help distinguish between possible scenarios driving FA. For instance, a good fit to a single statistical distribution may imply that the same process ope rates to create FA in all individuals in a population. In contrast, a good f it to a discrete mixture of several different density functions might suggest that highly asymme tric individuals suffe r from fundamentally different developmental processes than thei r more symmetric counterparts. Thin-tailed distributions (e.g., normal or exponential) ma y indicate relative independence in the accumulation of small random developmental e rrors, whereas heavy-tailed distributions may implicate non-independent cascades in the propagation of such error. Despite much interest in the relationship between envir onmental stress and levels of FA, the basic patterns of its distributio n in populations remain largely unexplored. One common distributional attr ibute of FA, leptokurtosis, has been discussed in the literature. Leptokurtosis denotes a distri bution that has many small and many extreme values, relative to the normal distribution. Two primary causes of this kind of departure from the normal distribution are the mixing of di stributions and/or sca ling effects in data. For example, the Laplace or double exponential distribution is leptokur tic (but not heavytailed) and can be represented as a continuous mixture of normal distributions (Kotz et al. 2001; Kozubowski and Podgorski 2001). Just as log scaling in the normal distribution results in the lognormal distribution, log scali ng in the Laplace leads to log-Laplace (also

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13 called double Pareto) distributions (Kozubow ski and Podgorski 2002; Reed 2001), which are both leptokurtic and heavytailed (see Figure 2-1). Several explanations for leptokurtosis in the distribution of FA have been proposed. Both i ndividual differences in developmental stability within a popul ation (Gangestad an d Thornhill 1999) and differences in FA between subpopulations (Houl e 2000) have been suggested to lead to continuous or discrete mixtur es of normal distributions with different developmental variances, which in turn would cause leptoku rtosis (e.g., a Laplace distribution) in the overall distribution of FA. Mixtures of nonnormal distributions may also cause either leptokurtosis or platykurtosis (more intermed iate values than the normal distribution: Palmer and Strobek 2003). A potential exampl e of this is illustrated by Hardersen and Framptons (2003) demonstration that a pos itive relationship between mortality and asymmetry can cause leptokurtosi s. Alternatively, Graham et al. (2003) have argued that developmental error should behave multiplicatively in actively growing tissues, creating a lognormal size distribution in most traits ra ther than the normal distribution that is usually assumed. They argue that this ultimately results in leptokurtosis (but not fat tails) and size dependent expression of FA. Because s imple growth models are often geometric, we should not be surprised if distributions of size-based FA followed the lognormal distribution (see Limpert et al 2001 for a review of lognormal distributions in sciences). Not well recognized within biology is the fact that close interaction of many components can result in powe r-law scaling (distributi onal tails that decrease proportional to x-a rather than to some exponential function of x such as exp(-ax) [exponential] or exp(-ax2) [normal] and hence to heavytailed distributions (Sornette 2003). Powerlaw scaling is often associated with the tail of the lognormal distribution,

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14 especially when log standard deviation is large ( Mitzenmacher unpublished ms.; Montroll and Shlesinger 1982, 1983; Roma n and Porto 2001; Ro meo et al. 2003). NORMAL LOGNORMAL LAPLACELOG-LAPLACE M I X T U R E M I X T U R ENormalLaplace Double Pareto lognormal (DPLN) 1 0 1 1 0 1 012345 y 202 5 additive/linear geometric/logarithmic Figure 2-1. Schematic representation of ma thematical relationships between candidate models for the distribution of fluctu ating asymmetry. Mixtures here are continuous. KEY: Solid line scale of variable (x ln(x)) Dashed line random walk observed at consta nt stopping rate (i.e., negative exponentially distributed stop times) Note: ra ndom walk on log scale exhibits geometric Brownian motion Dotted line convolution of two distributions (one of each type) Block arrow a continuous mixture of distributi ons with stochastic (exponentially distributed) variance (Note Log-Laplace is also called doubl e Pareto by Reed 2001, Reed and Jorgensen 2004)

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15 Because FA in a population may reflect fundamental developmental differences between different classes or groups of individuals, for exam ple stressed and non-stressed, or between different subpopul ations as suggested by Houl e (2000), we might expect discrete mixtures of different distributions to best describe FA. For instance, extreme individuals falling in a heavy upper tail may be those who have exceeded some developmental threshold. Major disruption of development, resulting in high FA, may also reveal the scaling that exists in the underlying gene regulatory netw ork (Albert and Barabasi 2002; Clipsham et al. 2002; Olvai an d Barabasi 2002). Alternatively, if FA is produced by a single process, but to various degrees in diffe rent individuals, then one might expect a continuous mixture model to best describe the di stribution of FA. The possibility of non-normal distribution of FA opens the door to several potential sampling problems. For instance, if the lognormal shape, or multiplicative variance parameter, is large, then broad distribution e ffects may slow the convergence of the sample mean to the population mean as sample sizes are increased (R omeo et al. 2003). An additional, thornier, problem is caused by power-law scaling in the tails of distributions. Many lognormally distributed datasets exhibit pow er-law scaling (or amplification) in th e tail region (Montroll and Shlesi nger 1982, 1983; Romeo et al. 2003; Sornette 2003), sometimes called ParetoLevy ta ils or just Levy tails. As sample sizes grow infinitely large, power-law and Pareto distributions may approach infinite mean (if the scaling exponent is less than three) and infinite variance (if the scaling exponent is less than two), and therefore will not obey the Law of Large Numbers (that sample means approach the population mean as sample sizes increase). Increased sampling actually increases the likelihood of samp ling a larger value in the tail of a Pareto distribution

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16 (Bardou et al. 2003; Quandt 1966), creating more uncertainty in estimates of the mean as sample size increases. The presence of pow er-law tails can slow overall convergence considerably even in distribut ions that are otherwise lognor mal with low variance (which may not look very different from well-beha ved lognormal distributions unless a large amount of data is accumulated). This discussion points to two effects that need to be assessed broad distribution effects, controlled by the shape (lognormal variance) parameter of the body of the FA distribution, and power-law tails, controlled by the scaling exponents of the tails of the FA distribution. To assess these effects, I apply a new statistical model, the double Pareto lognormal (DPLN) distribution (Reed and Jorg ensen 2004). The DPLN distribution is a lognormal dist ribution with power-law behavior in both tails (for values near zero and large positive values). Simila r to the log-Laplace di stributions, the DPLN distribution can be represente d as a continuous mixture of lognormal distributions with different variances. It can also be derived from a ge ometric Brownian motion (a multiplicative random walk) that is stopped or killed at a constant rate (i.e., the distribution of stop times is exponentially distributed: Reed and Jorgensen 2004; Sornette 2003). The parameters of the DPLN di stribution include a lognormal mean ( ) and variance ( 2) parameter which control the loca tion and spread of the body of the distribution, and power-law sca ling exponents for the left ( ) and right ( ) tails. Special cases of the DPLN include the right Pareto lognormal (RPLN) dist ribution, with a powerlaw tail on the right but not the left side ( ); the left Pareto lognormal (LPLN) distribution, with a power-law tail only near zero ( ); and the lognormal distribution, with no power-law tails ( ). For comparison, I also fit normal and half-normal

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17 distributions as well as the asymmetric La place distribution to the data on FA. See Figure 2-1 for schematic representation of re lationships between these candidate models for FA. In the following study, I directly test the f it of different distri butions to large FA datasets from three species of insects. I include a lab cultured monoclonal line of cotton aphid, Aphis gossipyii Glover, in an attempt to isolate the distribution of developmental noise for the first time. I also analyze da ta from a semi-wild population of domestic honeybee, Apis mellifera taken from a single inseminated single queen colony, and from a large sample of unrelated w ild-trapped Long-legged flies ( Chrysosoma crinitus : Dolichopididae). I address two primary groups of goals in this study. First, I investigate what distributions fit FA data best and how the parameters of th ese distributions vary across species, rearing conditions, and levels of genetic relatedness I also address whether outliers (individuals with visi ble developmental errors) appear to result from discrete or continuous processes. Secondly, I determin e how accurate the estimates of population mean FA are at various sample sizes to de termine whether past studies of FA been adequately sampled to accurately estimate mean or average FA in populations. In addition to these two primary goals, I also compare the best-fitting distributions and level of sampling error for three of the most co mmon methods of measuring FA: a univariate and a multivariate size-based metric of asymmetry, and a multivariate shape-based method. Methods Wings were collected and removed from three populations of insects and dry mounted on microscope slides. These populat ions included a m onoclonal population of

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18 1022 Cotton Aphid ( Aphis gossipyii Glover) started from a singl e individual collected from citrus in Lake Alfred, Florida, 1001 honeybees Apis mellifera maintained at University of Florida and 889 long-legged flies ( Chrysosoma crinitus :Dolichopodidae). All species identifications were made through the State of Florida Department of Plant Industry in Gainesville and voucher specimens remain available in their collections. Aphid cultures were maintained on potted plants in reach-in environmental chambers at 15C with constant 14/1 0 hour LD cycle generated by 4 20 watt fluorescent Grolux brand bulbs. Aphid cultures were cult ured on approximately 10 day old cotton seedlings ( Gossipium ) and allowed to propagate until crowded. Crowding stimulated alate formation (winged forms) in later generations which were collected every twenty days with a fine camel hair br ush wetted in ethanol. New plants were added every ten days and alates were allowed to move freely from plant to plant starting new clones until they were collected. The temperature at which the colony was maintained created a low temperature dark morph Co tton Aphid which still propagated on host plants quickly but was larger than high temperature light morphs that form at temperatures greater than 17C. Dark mo rphs colonize stems on cotton whereas light morphs colonized the undersides of leaves. The bees were collected in June 2004 by Dr. Glenn Hall at the University of Floridas Bee Lab from a single inseminated single queen colony. They were presumably all foragers and haplodiploid sist ers collected as they exited th e hive into a collection bag. The bag was frozen for three hours and then the bees were placed in 85% ethanol. Long-legged flies were trapped from a wild population using 14 yellow plastic water pan traps in southwest Gainesville, Florida, during May 2003 and May-June 2004.

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19 A very small amount of dishwashing deterg ent was added to the water to eliminate surface tension and enhance trapping. Traps were checked every three hours during daylight and set up fresh ever y day of trapping. All insect specimens were dried in 85% ethanol, and then pairs of wings were dissected (in ethanol) and air-dried to the gl ass slides while the ethanol evaporated. Permount was used to attach cover slips. This technique prevente d wings from floating up during mounting, which might slightly di stort the landmark configuration. Dry mounts were digitally photographed. All landmarks were identified as wing vein intersections on the digital images (six landmarks on each wing for aphids, eight for honeybees and Dolichopodid flies). See Appendi x A for landmark locations on wings for each species. Wing vein intersections were digitized three times each on all specimens using TPSDIG version 1.31 (Rohlf, 1999). All measures of FA were taken as the average FA value of the three replicate measurements for each specimen. Specimens damaged at or near any landmarks were disc arded. Fluctuating asymmetry was measured in three ways on all specimens. First, a common univariat e metric of absolute unsigned asymmetry was taken for two different landmarks: FA = abs(R L) where R and L are the Euclidean distances between the same two landmarks on either wing. In aphids, landmarks 1-2 and 2-3 were used; in bees, landmarks 1-4 and 2-6 were used; and in long-legged flies, landmarks 3-6 and 4-5 were used. Two multivariate geometric morphometrics using landmark-based methods were performed using all landmarks shown in Appendix A. A multivariate size-based FA (FA 1 in Palmer a nd Strobek 2003) was calculated as absolute value of (R L) where R and L are the centr oid sizes of each wing (i.e., the average of

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20 the distances of each landmark to their combin ed center of mass or cen troid location). In addition, a multivariate shape-based measure of FA known as the Procrustes distance was calculated as the square root of the sum of all squared Eucl idean distances between each left and right landmark after two-dimensional Procrustes fitting of the data (Bookstein 1991; Klingenberg and McIntyre 1998; FA 18 in Palmer and Strobeck 2003; Smith et al. 1997). Procrustes fitting is a three step process including a normalization for centroid size followed by superimposition of two sets of landmarks (right and left) and rotation until all distances between each landmark set is minimized. Centroid size calculation, Euclidean distance calculation and Procru stes fitting were performed using yvind Hammers Paleontological Statistics program PAST version 0.98 (Hammer 2002). For assessing measurement error (ME) of FA (or more specifically, the digitizing error), we conducted a Procrustes ANOVA (in Microsoft Excel) on all pairs of wing images resampled three times each for every species (Klingenberg and McIn tyre 1998). Percent measurement error was computed as (ME/average FA) x 100 where 3 / ) 3 1 3 2 2 1 ( FA FA FA FA FA FA ME All subsequent statistical analyses were performed using SPSS Base 8.0 st atistical software (SPSS Inc.). The fits of all measures of FA to eight distributional models (normal, half-normal, lognormal, asymmetric Laplace, double Pareto lognormal (DPLN), two limiting forms of DPLN, the right Pareto lognormal (RPLN) a nd the left Pareto lognormal (LPLN) and a discrete mixture of lognormal and Pareto) were compared by calculating negative log likelihoods and Likelihood Ratio Test (LRT ) if models were nested and Akaike Information Criteria (AIC) if not nested (Burnham and Anderson 1998; Hilborn and Mangel 1998). Both of these approaches penalize more complex models (those with

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21 more parameters) when selecting the best-fit distributional model for a given dataset. (The Likelihood Ratio Test doe s not technically apply when the nesting parameter is at the boundary of its allowed region, e.g., when for the DPLN, but Pinheiro and Bates (2000) suggest that the LRT is conser vative, favoring simpler models, under these conditions.) Best fitting parameters were obtained by maximizing the log-likelihood function for each model (Appendix B). Th e maximization was performed using the conjugate gradient method within unconstrai ned solve blocks in the program MathCad by MathSoft Engineering and Education Inc ( 2001), and was also confirmed using NelderMead simplex algorithm or quasi-Newt on methods in R version 2.0.1 (2003), a programming environment for data analysis and graphics. Phenodeviants were defined as individuals demonstrating missing wing veins, extra wing veins or partial wing veins on either one or both wings. All phenodeviants in honeybees involved absence of the vein at land mark 6 (LM 6). Phenodeviants in aphids were more variable but mostly involved abse nce of wing vein intersections at LM 2 or LM 3. Procrustes distances were estima ted for phenodeviants by omitting the missing landmarks (caused by the phenodeviance) and cont rolling for the effect of this removal on the sums of squares. I added an average of the remaining sums of squares in place of the missing sums of squares so that the calcu lated Procrustes distance is comparable to normal specimens (i.e., six landmarks). In almost all phenodeviants, this involved omission of only one set of landmark valu es. The frequency of phenodeviants was examined across the range of the FA distribut ion (i.e., Procrustes distance), and mean values of the FA for phenodeviants were co mpared to normal individuals in order to

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22 assess whether phenodeviants tended to show higher than normal levels of FA in characters that were not affected by the missing, partial, or extra wing veins. The best fitting parameters of the best fitting models were used to build a distributional model under which repeated sa mpling was simulated at various sample sizes. Average error in estimation of the m ean FA was calculated as a coefficient of variation ( 100 ) / ( x s) for 1,000 randomly generated datasets. Lastly, comparison were made of the estimation errors given the best fi tting distributions of FA to a dist ribution of sample sizes from 229 FA studies published in three recent meta-analyses (Polak et al. 2003; Thornhill and Mller 1998; Vollestad et al. 1999). Results In the distributions of shape-based FA in monoclonal cotton aphids (n = 1022), domesticated honeybees (n = 1001), and wild trapped long-legged flies (n = 889), AIC and LRTs always favored DPLN or RPLN m odels by a large margin. All size-based FA distributions favored DPLN or LPLN by a larg e margin (Table 2-1). All variants of discrete mixture models we tried had very poor results (data not shown). Figures 2-2 through 2-4 demonstrate best fitting models for multivar iate shape FA (DPLN and lognormal), multivariate centroid size FA (D PLN and half-normal), and univariate size FA (DPLN and half-normal) for all three spec ies. FA was often visually noticeable in aphids, where the mean shape FA (Procrus tes distance) was thre e times higher (0.062 0.00050) than in bees (0.023 0.00026) or flies (0.019 0.00028). I note that distribution of size FA in aphids and bees fit half-normal distribut ion in the upper tails fairly well but fit relatively poorly among i ndividuals with low FA. Long-legged flies exhibit poor fit to half-normal in both tails.

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23 Table 2-1. Maximized log-likelihood (MLL), number of model parameters (P) and Akaike Information Criterion differences ( AIC) for distributional models tested. Winning models have AIC zero. Models with goodness-of-fit nearly equal to winners are underscored ( AIC <3.0). 4.0 < AIC <7.0 indicates some support for specified model. AIC >10.0 indicates no support (Burnham and Anderson 1998). Distances between la ndmarks for first univariate size FA, aphids LM 1-2, bees LM 1-4 and long-legged fly LM 3-6 and for second univariate FA aphids LM 2-3, bees LM 2-6 and long-le gged fly LM 4-5. Species/model Multivariate shape FA Multivariate size FA First univariate size FA Second univariate size FA Cotton Aphid P MLL AIC MLL AIC MLL AIC MLL AIC DPLN 4 255.117 0.000 542.906 1.236 545.725 2.815 614.123 2.333 RPLN 3 264.72 17.207 547.44 8.303 549.789 8.944 617.498 7.083 LPLN 3 256.723 1.213 543.288 0.000 545.317 0.000 613.957 0.000 LNORM 2 734.726 955.218 1261 1433 1265 1438 1413 1596 NORM 2 2241 3968 2847 4606 3352 5612 3534 5838 HNORM 1 3523 6530 2370 3650 2893 4691 3047 4862 LAPLACE 3 637.044 761.855 1638 2189 1883 2676 1950 2671 Honey Bee DPLN 4 132.908 7.369 536.278 2.744 454.337 0.000 515.93 6.329 RPLN 3 130.224 0.000 549.641 27.47 460.51 10.346 525.675 23.82 LPLN 3 136.361 12.275 535.906 0.000 455.483 0.292 513.765 0.000 LNORM 2 523.667 784.888 1262 1451 1081 1249 1211 1393 NORM 2 3371 6480 2180 3286 1709 2505 2302 3574 HNORM 1 5319 10370 1775 2475 2687 1771 1924 2816 LAPLACE 3 1148 2036 1387 1703 1162 1413 1445 1863 Long-legged Fly DPLN 4 146.603 37.971 452.728 0.000 453.958 0.000 412.007 0.000 RPLN 3 128.617 0.000 454.056 0.656 460.261 10.606 422.993 20.182 LPLN 3 149.585 41.935 455.818 4.18 456.237 2.559 412.902 0.212 LNORM 2 506.553 753.871 1055 1201 1064 1216 985.894 1144 NORM 2 2985 5710 1760 2610 1806 2701 1668 2508 HNORM 1 5147 10030 1084 1256 1252 1590 1099 1368 LAPLACE 3 1041 1825 940.241 973.02 1016 1121 997.28 1169 For multivariate shape analysis, right tail power-law exponents ( ) were very high (thousands), left-t ail exponents ( ) from 3.9-9.9, while the dispersion parameter ( ) was narrowly distributed from 0.310 to 0.356. Thus, shap e FA exhibited little scaling in tails (i.e., nearly lognormal). For size-based FA, dispersion wa s much larger (0.57-0.74), and power-law exponents more variable for univari ate and multivariate size-based FA (left

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24 tail ( ) and right tail ( ) were generally low, indicating moderate power-law scaling in both tails) as shown in Table 2-2. Table 2-2. Best fit parameters for models in Table 1. Parameters for univariate size FA are very similar to multivariate size FA and are not shown. Skew indexes for asymmetric Laplace are also not shown. Multivariate shape FA Multivariate size FA Cotton Aphid Location Dispersion/ Shape Right tail Left Tail Location Dispersion/ Shape Right tail Left Tail DPLN -2.62 0.356 1160 4.04 1.45 0.735 6.24 3.01 RPLN -3.01 0.415 7.03 1.07 0.800 4.58 LPLN -2.62 0.353 3.90 1.73 0.699 2.23 LNORM -2.87 0.434 1.28 0.824 NORM 0.062 0.027 4.92 3.94 HNORM 0.009 0.059 0.163 6.31 LAPLACE 0.049 0.058 0.657 2.19 Honey Bee DPLN -3.74 0.310 8380 9.80 1.32 0.565 10.4 1.57 RPLN -4.02 0.274 5.52 0.571 0.829 4.82 LPLN -3.71 0.305 7.75 1.52 0.510 1.35 LNORM -3.84 0.327 0.777 0.850 NORM 0.023 0.008 2.95 2.14 HNORM 0.007 0.018 0.078 3.64 LAPLACE 0.021 0.018 0.550 2.24 Long-legged Fly DPLN -3.93 0.346 4940 10.3 2 -0.067 0.683 3.09 5.14 RPLN -4.28 0.231 3.73 -0.302 0.698 2.75 LPLN -3.92 0.341 9.91 2 0.319 0.743 3.85 LNORM -4.02 0.351 0.060 0.783 NORM 0.019 0.008 2.00 2.24 HNORM 0.007 0.015 0.075 2.11 LAPLACE 0.017 0.017 0.346 1.06 Figure 2-5 shows the distributi on of FA sample sizes from 229 studies published in three recent meta-analyses (Polak et al. 2003; Thornhill and Mller 1998; Vollestad et al. 1999).

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25 00.020.040.060.080.10.120.140.160.180.20.220.24 0 5 10 15 20 25 30 35 00.050.10.150.20.25 0 20 40 60 expDPLN obsDATAA. Cotton Aphid 00.050.10.150.20.25 0 0.1 0.2 0.3 expLGN obsDATA 00.010.020.030.040.050.060.070.080.09 0 10 20 30 40 50 60 70 80 90 00.020.040.060.08 0 5 10 15 expDPLN obsDATAB. Domestic Honeybee 00.020.040.060.08 0 0.05 0.1 expLGN obsDATA 00.010.020.030.040.050.060.070.08 0 20 40 60 80 100 00.020.040.060.08 0 5 10 expDPLN obsDATAC. Long-legged Fly 00.020.040.060.08 0 0.02 0.04 0.06 expLGN obsDATADPLN DPLN DPLN Lognormal Lognormal Lognormal Figure 2-2. Distribution of multivaria te shape FA of A) cotton aphid ( Aphis gossipyii ) B) domestic honeybee ( Apis mellifera ) and C) long-legged fly ( Chrysosoma crinitus ). Best fitting lognormal (dashed line, lower inset), and double Pareto lognormal (solid line, uppe r inset) are indicated.

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26 02468101214161820222426 0 5 10 15 20 25 30 35 40 45 50 55 60 051015 10 expDPLN obsDATAA. Cotton Aphid 051015 0 10 expHNORM obsDATA 012345678910111213 0 5 10 15 20 25 30 35 40 45 50 0510 20 40 expDPLN obsDATAB. Domestic Honeybee 0510 0 5 10 expHNORM obsDATA 012345678 0 10 20 30 40 50 60 70 80 90 100 110 120 0246810 5 10 expDPLN obsDATAC. Long-legged Fly 0246810 0 5 10 expHNORM obsDATADPLN DPLN DPLN Half-normal Half-normal Half-normal Figure 2-3. Distribution of multivariate centroid size FA of A) cotton aphid ( Aphis gossipyii ) B) domestic honeybee ( Apis mellifera ) and C) long-legged fly ( Chrysosoma crinitus ). Best fitting half-normal (d ashed line, lower inset) and double Pareto lognormal distribution (solid line, upper inse t) are indicated.

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27 0510152025303540 0 5 10 15 20 25 30 35 40 45 50 55 60 010203040 20 40 expDPLN obsDATAA. Cotton Aphid (LM 1-2) 010203040 0 20 40 expHNORM obsDATA 051015202530354045 0 5 10 15 20 25 30 35 40 45 50 55 60 010203040 20 40 expDPLN obsDATAA. Cotton Aphid (LM 2-3) 010203040 0 20 40 expHNORM obsDATA 0123456789 0 5 10 15 20 25 30 35 40 45 50 55 60 02468 5 expDPLN obsDATAB. Domestic Honeybee (LM 1-4) 02468 0 5 expHNORM obsDATA 0 1 2 3 4 5 6 7 8 9 10 11 12 0 5 10 15 20 25 30 35 40 45 02468 5 10 expDPLN obsDATAB. Domestic Honeybee (LM 2-6) 0510 0 5 10 expHNORM obsDATA 012345678910 0 10 20 30 40 50 60 70 80 90 100 110 120 n 0246810 5 10 expDPLN obsDATAC. Long-legged Fly (LM 3-6) 0246810 0 5 10 expHNORM obsDATA 012345678910 0 5 10 15 20 25 30 35 40 45 50 55 60 0246810 5 10 expDPLN obsDATAC. Long-legged Fly (LM 4-5) 0246810 0 5 10 expHNORM obsDATADPLN Half-normal Half-normal Half-normal Half-normal Half-normal Half-normal DPLN DPLN DPLN DPLN DPLN Figure 2-4. Distribution of univariate unsigned size FA of A) cotton aphid ( Aphis gossipyii ) B) domestic honeybee ( Apis mellifera ) and C) long-legged fly ( Chrysosoma crinitus ). Best fitting half-normal (dashed line, lower inset) and double Pareto lognormal distribution (solid line, upper inset) are indicated.

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28 sample or treatment size4 50 4 00 3 50 3 00 2 5 0 200 1 50 1 00 50 5frequency40 30 20 10 0 Figure 2-5. Distribution of sample sizes (n) from 229 fluctuating asymmetry studies reported in three recent meta-analyses (Vollestad et al. 1999, Thornhill and Mller 1998 and Polak et al. 2003). Only five studies had sample sizes greater than 500 (not shown). Nearly 70% of the 229 FA studies have sa mple or treatment sizes less than 100. Figure 2-6 demonstrates the hypot hetical error levels (coeffi cients of variations) in estimated mean FA at various sample sizes. Approximate best fit parameters were used to estimate the coefficient of variation (C V) under the DPLN distribution (for shape FA, = -3.7, = 0.35, = 1000, = 9; for size FA, = 1.2, = 0.70, = 4.0, = 4.0). For the same set of landmarks, multivariate shape FA measures lead to the least amount of error in estimating mean FA under DPLN at any sample size. Both univariate and multivariate size FA perform more poorly in terms of both convergence and overall percentage error. I found that while phenodeviants occurred in almost all regions of the distribution range of FA, the percentage of phenodeviant individuals increased dramatically with

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29 increasing FA (Figure 2-7; aphids, r = 0.625 p = 0.013 ; bees, r = 0.843 p = 0.001). I also found that individuals with phenodeviant wings (both aphids and bees) showed significantly higher levels of FA across those wing landmarks unaffected by the phenodeviant traits (p < 0.002 in both aphids and bees). Only a single case of phenodeviance was sampled in long-legged flies. 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25 30 35 40 45 Legend shape FA size FA SAMPLE SIZE % ERROR IN ESTIMATION OF THE MEAN (CV) Figure 2-6. Relationship between sample si ze and % error for estimates of mean FA drawn from best fitting size (dashed line ) and shape (solid line) distributions using 1000 draws per sample size. All runs use typical winning double Pareto lognormal parameters (shape FA = -3.7, = 0.2, = 1000, = 9; for size FA = 1.2, = 0.7, = 4.0, = 4.0). Percent measurement error for shape FA was 1.41% in aphids, 1.63% in bees, and 2.42 % in flies while for size FA, it was ~4.5% in aphids, ~5% in bees, and ~6.5 % in flies. In a Procrustes ANOVA (Klingenberg and McIntyre 1998) the mean squares for the interaction term of the ANOVA (MSInteraction) was highly significant p<0.001 in all three species indicating that FA variation was significantly larger than variation in measurement error (ME). The distribution of signed ME was normal and exhibited

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30 moderate platykurtosis for all types of FA in all species examined. Measurement error was very weakly correlated to FA in all samples (0.01 < r2 <0.07). 125 1070 N =APHID WINGSPHENODEVIANT NORMAL WINGMean +1 SE Procrustes Distance.064 .062 .060 .058 .056 .054 .052 65 1035 N =BEE WINGSPHENODEVIANT NORMAL WINGMean +1 SE Procrustes Distance.030 .029 .028 .027 .026 .025 .024 .023 C D 0 t o 0 0 1 0 0 1 t o 0 0 2 0 0 2 t o 0 0 3 0 0 3 t o 0 0 4 0 0 4 t o 0 0 5 0 0 5 t o 0 0 6 0 0 6 t o 0 0 7 0 0 7 t o 0 0 8 0 0 8 t o 0 0 9 0 0 9 t o 0 1 0 1 t o 0 1 1 0 1 1 t o 0 1 2 0 1 2 t o 0 1 3 0 1 3 t o 0 1 4 0 1 4 t o 0 1 5 0 1 5 t o 0 1 6 0 1 6 t o 0 1 7 0 1 7 t o 0 1 8 0 1 8 t o 0 1 9 0 1 9 to 0 2 0 50 100 150 200 250 300 Legend phenodeviants normal Procrustes Distance FrequencyA 0 t o 0 0 0 4 5 0 0 0 4 5 t o 0 0 0 9 0 0 0 9 t o 0 0 1 3 5 0 0 1 3 5 t o 0 0 1 8 0 0 1 8 to 0 0 2 2 5 0 0 2 2 5 to 0 0 2 7 0 0 2 7 to 0 0 3 1 5 0 0 3 1 5 t o 0 0 3 6 0 0 3 6 to 0 0 4 0 5 0 0 4 0 5 t o 0 0 4 5 0 0 4 5 t o 0 0 4 9 5 0 0 4 9 5 t o 0 0 5 4 0 0 5 4 t o 0 0 5 8 5 0 0 5 8 5 t o 0 0 6 3 0 0 6 3 t o 0 0 6 7 5 0 0 6 7 5 t o 0 0 7 2 0 0 7 2 t o 0 0 7 6 5 0 0 7 6 5 t o 0 0 8 1 0 0 8 1 t o 0 0 8 5 5 0 0 8 5 5 t o 0 0 9 0 50 100 150 200 250 300 Legend phenodeviants normal Procrustes DistanceB Figure 2-7. The proportion and percentage (inset) of individuals with visible developmental errors on wings (phenode viants) are shown for cotton aphids (A) and honeybees (B) in relation to distribution of shape FA (Procrustes distance). Average FA for both nor mal and phenodeviant aphids (C) and honeybees (D) are also given. Discussion The Distribution of FA The data demonstrate a comm on pattern of distribution in the FA in wing size and shape of three different species whose populations existed under very different environmental conditions (lab culture, free-li ving domesticated, and wild) and genetic structure (monoclonal, haplodipl oid sisters, and unrelated). The similarities across the very different species and r earing conditions used in this study suggest that the

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31 distribution of size and shape FA ma y have universal parameters (e.g., 0.35 for shape FA and 0.7 for size based FA). The data co nfirm that although size FA sometimes exhibits reasonable fit to half-normal in the upper tail, a nd shape FA is reasonably well fit by lognormal distributions, large datasets of FA in both size and shape are always best described by a double Pareto lognormal distribut ion (DPLN) or one of its limiting forms, LPLN and RPLN. Multivariate shape FA demo nstrates narrow distri bution with a large right tail, including the top fe w percent of the most extremely asymmetric individuals, that is best fit by DPLN or RPLN. Both univariate and multivariate size FA exhibit a considerably broader distribution with moderate leptokurtosis that is best fit by DPLN or LPLN. The data suggest that the DPLN di stribution and its limiting forms are generally the most appropriate models for the dist ribution of FA regardless of method of measurement. Evidence that distribution of FA clos ely follows DPLN, a continuous mixture model, and appearance of phenodeviance across n early the entire range of data suggested that developmental errors may be cause d by a similar process across the entire distribution of FA in a populat ion. In other words, variati on in FA may have a single cause in most of the data. Although phenode viance is significantly related to increased levels of FA and is more prevalent in the ri ght tail region of the shape FA distribution, it does not appear associated exclusively with th e right tail, as a thre shold model for high FA might predict. The very poor fits to all variations on the discrete mixture model also suggest a lack of distinct pro cesses creating extreme FA in th e three datasets. However, I caution that use of maximum like lihood methods to fit data to discrete mixtures is often technically challenging. I found no block e ffects (e.g., no differences in FA levels

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32 between long-legged fly samples collected on different weeks or aphid samples collected from different pots or growth chambers), so it appears there is no obvious sample heterogeneity that could result in a discre te mixture. With the usual caveats about inferring process from pattern, I do not find obvious thresholds in the distribution of asymmetries at the population level that would suggest threshold eff ects at a genetic or molecular level. Lastly, based on the appearance of only one phenodeviant among our wild trapped long-legged fly population (as oppos ed to many in the bee and lab reared aphid populations), I speculate that mortality related to phenodeviance (and perhaps high FA) in wing morphology may be relaxed in lab cu lture and domestication. But this could be confounded by other differences between the three datasets including genetic redundancy and species differences. Furthe r comparisons among populations of single species under different conditions woul d be needed to test this idea. Sample Size and the Estimation of Mean FA In random sampling under DPLN, we found th at broad distributi on effects due to the shape parameter were minimal in thei r effect of slowing convergence to the population mean in multivariate shape-based FA ( 0.35 for all three datasets). However, these effects are considerable for univariate and multivariate size-based FA (where 0.70). The effects of scaling in th e tails of the distribution, which cause divergence from the mean, appear to have little effect in th e right tail of the distribution of shape-based FA. However, larger effect s in the tails of the size FA create more individuals with very low a nd very high asymmetry than expected under the assumption of normality. The point estimates of the sc aling exponents for size FA are close to the range where very extreme values may be sampled under the di stribution tails (if < 3 or <3), greatly affecting confidence in the estim ate of mean FA. With a sample size of 50

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33 and the best fitting DPLN parameters typica l for asymmetry in multivariate shape, we found that the coefficient of variation for m ean FA is about 5%, whereas size-based mean FA fluctuates about 13% from sample to sample. At a sample size of 100, these coefficients of variation ar e 3.2% and 7.5% respectively. Un less experimental treatment effects in most FA studies are larger than this, which is unlikely in studies using sizebased measures of FA of more canalized trai ts, statistical power and repeatability will be low. Given the sample size range of most previous studies (n = 30-100) and their tendency to favor size-based measurement me thods, our results suggest that many past FA studies may be under-sampled. Furthermore, it is also likely that given the small sample sizes in many FA studies, particularly involving vertebrate s where n < 50, the tail regions of natural FA distributions are often severely under sampled and sometimes truncated by the removal of outliers. These factors may artificially cause non-normal distributions to appear normal, also potential ly resulting in inaccurate estimation of mean FA. The Basis of Fluctuating Asymmetry The surprisingly good fit of FA distribu tions to the DPLN model in our study suggests that the physical basis of FA ma y be created by the combination of random effects in geometrically expa nding populations of cells on either side of the axis of symmetry (i.e., geometric Browni an motion). Studies in the Drosophila wing indicate that cell lines generally compete to fill a prescribed space during development with more rapidly dividing lines out-competing weaker ones (Day and Lawrence 2000). Because regulation of the growth of such cell popul ations involves either nutrients and/or signaling substances that stop the cell cycl e when exhausted, it is likely that the distribution of numbers of cells present at the completion of growth follows an

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34 exponential distribution. Reed and Jorgense n (2004) demonstrate that when a population of repeated geometric random walks is ki lled at such a constant rate, the DPLN distribution is the natural result. There are many other examples of growth processes in econometrics and physics where random pr oportional change combined with random stopping/observation create size distributions of the kind described here (Reed 2001; Kozubowski and Podgorski 2002). In the future, when applying this model to instability during biological growth, it would be very interesting to inves tigate how genetic and environmental stress might affect the parameters of this model. If scaling effects are found to vary with stress, then leptokurtosis may potentially be a be tter candidate signal of developmental instability than increased mean FA. Conclusion Although size-based FA distributions can sometimes appear to fit normal distributions reasonably well as previous definitions of FA suppose, I demonstrate that three large empirical datasets all support a ne w statistical model for the distribution of FA (the double Pareto lognormal distribution), which potentially exhibits power-law scaling in the tail regions and leadi ng to uncertain estimation of tr ue population mean at sample sizes reported by most FA studies. The assumption of normality fails every time candidate models are compared on large data sets. Failure of this assumption in many datasets may have been a major source of disc ontinuity in results of past FA studies. Future work should attempt to collec t larger sample/treatment sizes (n 500) unless the magnitude of treatment effects on FA (and thus the statistical power of comparisons) is very large. Our results demonstrate that multivariate shape-based methods (Klingenberg and McIntyre 1998) result in more repeatab le estimates of mean FA than either multivariate or univariate size-based me thods. I would also recommend that

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35 methodology be re-examined even in large sample studies of FA. For example, because Drosophila are usually reared in ma ny replicates of small tube s with less than 50 larvae per tube, many large studies may still be co mpromised by individual sizes of replicate samples. I also suggest that authors of pa st meta-analyses and reviews of FA literature reassess their conclusions after excluding studi es in which under-sampling is found to be problematic. Careful attention to distributional and sampling i ssues in FA studies has the potential both to mitigate problems with repeatability and possibly to suggest some of the underlying mechanisms driving variation in FA among individuals, populations, and species.

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36 CHAPTER 3 INBREEDING REDUCES POWER-LAW SC ALING IN THE DISTRIBUTION OF FLUCTUATING ASYMMETRY: AN EX PLANATION OF THE BASIS OF DEVELOPMENTAL INSTABILITY Introduction Fluctuating asymmetry (FA) is the average difference in size or shape of paired or bilaterally symmetric morphol ogical trait sampled across a population. The study of FA, thought to be a measure of developmental instability, has a controversial history. Fluctuating asymmetry is hypothesized by some to universally represent a populations response to environmental and/or genetic stress (Parsons 1992, Clarke 1993, Graham 1992). It is also generally accepted that FA ma y be co-opted as an indicator or even a signal of individual genetic buffering cap acity to environmen tal stress (Moller 1990, Moller and Pomiankowski 1993). R ecent literature reviews reveal that these conclusions are perhaps premature and anal yses of individual studies often demonstrate conflicting results (Lens 2002, Bjorksten 2000). Babbitt et al (2006) demonstrate that this conflict may be caused by under-sampling due to a fals e assumption that FA always exhibits a normal distribution. Also, FA may be responding to experimental treatment in a complex and as yet unpredictable fashion. Until th e basis of FA is better understood, general interpretation of FA studies remains difficult. What is the Basis of FA? Earlier studies of sexual selection and FA conclude that FA is ultimately a result of strong selection against th e regulation of the developmen t of a particular morphology (e.g., morphology used in sexual display). T hus in some instances FA may increase and

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37 therefore become a better signal and/or more honest indicator of good genes. Sexually selected traits tend to have increased FA (Moller and Swaddle 1997), however the exact mechanism by which FA increases remains unexplained (i.e., a black box). More recently, theoretical attempts have been made to explain how FA may be generated but none have been explicitly tested. Models for the phenomenological basis of FA fall into two general categories: reaction-diffusion models and diffusion-threshold models (reviewed in Klingenberg 2003). The former class of models involves the chaotic and nonlinear dynamics in the regulation or negative feedback among neighboring cells (Emlen et al. 1993) or adjacent bilateral morphology (Graham et al. 1993). The latter class of models combines morphogen diffusion and a threshold response, the parameters of which are controlled by hypothetical genes and a small amount of random developmental noise (Klingenberg and Nijhout 1 999). The result of this latter class of model is that different genotypes respond di fferently to the same amount of noise, providing an explanation for gene tic variation in FA response to the same environments. Traditionally, models for the basis of FA assume that variation in FA arises from independent stochastic events that influence the regulati on of growth through negative feedback rather than processes that may fuel or promote growth. None of these models explicitly or mathematically address the effect of stochastic behavior in cell cycling on the exponential growth curve. More recently Graham et al. (2003) makes a compelling argument that fluctuating asymmetry often results from multiplicative errors during growth. This is consistent with one part icular detail about ho w cells behave during growth. For several decades there has been evidence that during development, cells actually compete to fill prescribed space unt il limiting nutrients or growth signals are

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38 depleted (Diaz and Moreno 2005, Day and La wrence 2000). Cell populations effectively double each generation until signaled or forced to stop. It has also been observed that in Drosophila wing disc development, that synchro ny in cell cycling does not occur across large tissue fields but rather extends only to an average cl uster of 4-8 neighboring cells regardless of the size and stage of development of the imaginal disc (Milan et al. 1995). The assumption of previous models, particul arly the reaction-diffusion type, that cell populations are collectively controlling their cell cyc ling rates across a whole developmental compartment is probably unreali stic. Regulatory control of fluctuating asymmetry almost certainly does occur, but probably at a higher level involving multiple developmental compartments where competing cells are prevented from crossing boundaries. However, given that indivi dual cells are behaving more or less autonomously during growth within a single de velopmental compartment, I suggest that variation in fluctuating asy mmetry can be easily generated at this level by a process related to stochastic exponen tial expansion and its termination in addition to regulatory interactions that probably act at higher levels in the organism. In this paper, I explore and test simple model predictions regarding th e generation of fluctuating asymmetry though multiplicative error without regulatory feedback. Exponential Growth and Non-No rmal Distribution of FA In previous work, Babbitt et al. (2006) de monstrate that the distribution of unsigned FA best fits a lognormal distribution with scaled or power-law tails (double Pareto lognormal distribution or DPLN). This distribution can be generated by random proportional (exponential) growth (or geometri c Brownian motion) that is stopped or observed randomly according to a negative expo nential probability (Reed and Jorgensen 2004). I suggest that this source of power-law scaling in the ta ils of the FA distribution is

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39 also the cause of leptokurtosis th at is often observed empirically in the distribution of FA. Kurtosis is the value of the standardized four th central moment. Like the other moments, (location, scale, and skewness), kurtosis is best viewed as a concept that can be formalized in multiple ways (Mosteller and Tukey 1977). Leptokurtosis is best visualized as the location and scale-free movement of probability mass from the shoulders of a symmetric distribution towa rds both its center and tail (Balanda and MacGillivray 1988). Both the Pareto and power-function distributions have shapes characterized by the power-law and a larg e tail and therefore exhibit a lack of characteristic scale. Both because kurtosis is strongly affected by tail behavior, and because leptokurtosis involves a diminishing of characteristic scal e in the shape of a distribution, the concepts of scaling and kurtosis in real data can be, but are not necessarily always, inte r-related. Both in the past and very recently, leptokur tosis in the distribution of FA has been attributed to a mixture of normal FA distri butions caused by a mixi ng of individuals, all with different genetically-based developmen tal buffering capacity, or in other words, different propensity for expressing FA (Gangestad and Thornhill 1999, Palmer and Strobek 2003, Van Dongen et al. 2005). Alt hough not noted by these authors, continuous mixtures of normal distributions generate the Laplace distribution (Kotz et al. 2001, Kozubowski and Podgorski 2001) and can be distinguished from other potential candidate distributions by using appropriate model selection techniques, such as the Akaike Information Criterion technique (Burnham and Anderson 1998). Graham et al. (2003) rejects the typical e xplanation of leptokurtosis th rough the mixing of normal distributions by noting that differences in random lognormal variable can generate

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40 leptokurtosis. Babbitt et al. (2006) also reject the explanation that leptokurtosis in the distribution of FA is caused by a mixture of normal distributions because the double Pareto-lognormal distribution, not the Laplace di stribution, always appears the better fit to large samples of FA. Therefore, leptokur tosis, often observed in the distribution of FA, may not be due to a mixing process, but in stead may be an artifact of scaling in the distribution tail, which pr ovides evidence of geometric Brownian motion during exponential expansion of populations of cells. Testing a Model for the Basis of FA I propose that the proximate basis for variat ion of FA in a population of organisms is due to the random termination of stochas tic geometric growth. The combination of opposing stochastic exponential functions re sults in the slow power-law decay that describes the shape of the distributions tail. In this paper, I present a model for FA and through simulation, test the prediction that ge netic variation in the ability to precisely terminate growth will lead to increased kur tosis and decreased s caling exponent in the upper tail of the distribution. Then, I asse ss the validity of this model by direct comparison to the distribution of FA within large samples of wild and inbred populations of Drosophila Under the assumption that a less heterozygous population will have less variance in the termination of growth, I w ould predict that inbr eeding should act to reduce power-law scaling effects in the distribution of FA in a population. Inbreeding should also reduce the tail we ight (kurtosis) and mean FA assuming inbred individuals have lower variance in the times at which th ey terminate growth. I also assume that inbreeding within specific lines does not act to amplify FA due to inbreeding depression. It has been demonstrated that Drosophila melanogaster do not increase mean FA in response to inbreeding (Fowler and Whitlock 1994) and it is suggested that large

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41 panmictic populations typical of Drosophila melanogaster may not harbor as many hidden deleterious recessive mutations as other species (Houle 1989), making them resistant to much of the typi cal genetic stress of inbreeding. Therefore, the absence of specific gene effects duri ng inbreeding suggests that Drosophila may be a good model for investigating the validity of our model as an explana tion of the natural variation occurring in population level FA. Methods Model Development Simulation of geometric Brownian motion Ordinary Brownian motion is most easily simulated by summing independent Gaussian distributed random numbers or white noise (iX ). See Figure 3-1. (3.1) n i iX X W1) ( which simulated in discrete steps is (3.2) 1 1 t t tW N N where N = cell population size, t = time st ep and W = a random Gaussian variable. Exponential or geometric Brownian motion, a ra ndom walk on a natural log scale, can be similarly simulated. Geometric Brownian motion is described by the stochastic differential equation (3.3) ) ( ) ( ) ( ) ( t dW t Y dt t Y t dY or also as (3.4) ) ( ) ( ) ( t dW dt t Y t dY

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42 ORDINARY BROWNIAN MOTION-600 -400 -200 0 200 400 600 800 1000 124477093116139162185208231254277300323346369392415438461484 WHITE NOISE-150 -100 -50 0 50 100 150 121416181101121141161181201221241261281301321341361381401421441461481 timeW N Figure 3-1. Ordinary Brownian motion (l ower panel) in N simulated by summing independent uniform random variables (W) (upper panel). where W(t) is a Brownian motion (or Weiner process) and and are constants that represent drift and volatility respectively. Equation 3.3 has a lognormal analytic solution (3.5) ) ( ) 2 (2) 0 ( ) (t W te Y t Y A simulation of geometric Brownian mo tion in discrete form follows as (3.6) 1 1 1 t t t tW N N N where N = cell population size, t = time step and W = a random Gaussian variable. See Figure 3-2. Equation 3.6 is iden tical to the equation for multip licative error in Graham et al. (2003). I modify equa tion 3.6 slightly by letting W range uniformly from 0.0 to 1.0 with W= 0.5 and adding the drift constant C that allows for stocha stic upward drift (at C > 0.5) or downward drift (at C < 0.5).

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43 (3.7) 1 1 1 1 1) ( t t t t t tN W C W N CN N At C = 0.5, eqns. 3.6 and 3.7 behave identical ly. Geometric Brownian motion with upward drift is shown in Figure 3-3. 0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 13569103137171205239273307341375409443477 time step 0.0000001 0.000001 0.00001 0.0001 0.001 0.01 0.1 1 13671106141176211246281316351386421456 time step Log N Ntime Figure 3-2. Geometric Brownian motion in N and log N simulated by multiplying independent uniform random variables. This was generated using Equation 1.5 with C = 0.54.

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44 0 500000000 1000000000 1500000000 2000000000 2500000000 3000000000 3500000000 4000000000 4500000000 13773109145181217253289325361397433469 time step 0.00001 0.001 0.1 10 1000 100000 1000000 0 1E+09 1E+11 13467100133166199232265298331364397430463 time step Log N Ntime Figure 3-3. Geometric Brownian motion in N and log N with upward drift. This was generated using Equation 1.5 with C = 0.60. Simulation of fluctuating asymmetry Using the MathCad 13 (Mathsoft En gineering and Education 2005), two independent geometric random walks were performed and stopped randomly at mean time t= 200 steps with some variable normal probabilit y. The random walks result in cell population size equal to tN(orL tN and R tN on left and right sides respectively). Fluctuating asymmetry (FA) was defined as the difference in size resulting from this random proportional growth on both sides of th e bodies axis of symmetry plus a small degree of random uniformly distributed noise or

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45 (3.8) rv N N FAR t L t where (rv) was uniformly distribute d with a range of .1(R t L tN N ). Using a MathCad-based simulation in VisSim LE, the generation of individual FA values was repeated until a sample size of 5000 was reached. The random noise (rv) has no effect on the shape of the FA distribution but fills empty bins (gaps) in distri bution tails. Because rv is small in comparison toR t L tN N its effect is similar to that of measurement error (which would be normally distributed rather th an uniform). Schematic representation of the simulation process for Reed and Jo rgensen (2004) model and simulation of fluctuating asymmetry are shown in Figure 4 an d 5. The simulation of the distribution of fluctuating asymmetry was compared at norma l standard deviation of termination of growth (t) ranging from = 0.5, 0.8, 1.2,3 and 7 with a drift constant of C = 0.7. Inbreeding Experiment In May 2004 in Gainesville, Florida, 320 free-living Drosophila simulans were collected in banana baited traps and put into a large glass jar and cultured on instant rice meal and brewers yeast. After two genera tional cycles 1000 indivi duals were collected in alcohol. This was repeated again in June 2005 with 200 wild trapped flies. Lines of inbred flies were created from the May 2004 wild population through eight generations of full sib crosses removing an estimated 75% of the preexisting heterozygosity (after Crow and Kimura 1970) Initially, ten in dividual pairs were isolated from the stock culture and mated in pint mason jars with media and capped with coffee filters. In each generation, and in each line, and to ensure that inbred lines were not accidentally lost t hough an inviable pairing, four pairs of F1 sibs from each cross were then mated in pint jars. Offs pring from one of these four crosses were

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46 randomly selected to set up the next genera tion. Of the original ten lines, only four remained viable after eight generations of full sib crossing. These remaining lines were allowed to increase to 1000+ individuals in 1 quart mason jars and then were collected for analysis in 85% ethanol. This generall y took about 4 generations (8 weeks) of open breeding. One completely isogenic (balancer) line of Drosophila melanogaster was obtained from Dr. Marta Wayne, University of Florida and also propagated and collected. GROWTH STOP TIME NEGATIVE EXPONENTIAL VARIATION IN GROWTH STOP TIME 0100200300400500 0 50 100 150 200 0 1 e lDOUBLE PARETO LOGNORMAL DISTRIBUTION REPEAT SAMPLING 0100 0 2 0 10 20 30 40 50 60 70 80 90 100 1E-006 1E-005 0.0001 0.001 0.01 RANDOM WALKS WITH GEOMETRIC BROWNIAN MOTION Figure 3-4. Model representation of Reed and Jorgensens (2004) physical size distribution model. Variable nega tive exponentially distributed stopping Times of random proportional growth (GBM with C = 0.5) create double Pareto lognormal distribution of size.

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47 Figure 3-5. A model representa tion of developmental instab ility. Normally variable stopping times of random proportional gr owth (GBM with C > 0.5) create double Pareto lognormal distribution of size. All flies were cultured on rice meal a nd yeast at 29C with 12:12 LD cycle in environmentally controlled chambers at th e Department of Entomology and Nematology at the University of Florida. Wings were collected and removed from 1000 flies from each of the two samples of wild population a nd four samples of inbred lines and dry mounted on microscope slides. Specimens were dried in 85% ethanol, and then pairs of Stochastic Geometric Growth1 10000 1E+08 1E+12 1E+16 1E+20 1E+24 1E+28 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748log Cell Number 050100150 0 2 4 GROWTH STOP TIME NORMAL VARIATION IN GROWTH STOP TIME RANDOM WALKS WITH GEOMETRIC BROWNIAN MOTIONSIZE DISTRIBUTION REPEAT SAMPLING 01002003004005 00 0 50 100 150 200

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48 wings were dissected (i n ethanol) and air-dried to the gla ss slides. Permount was used to attach cover slips. This technique prev ented wings from floating up during mounting, which might slightly distort the landmark configuration. Dry mounts were digitally photographed. All landmarks were identified as wing vein intersections on the digital images (eight landmarks on each wing). See Appendix A for landmark locations. Morphometric analyses Wing vein intersections were digitized on all specimens using TPSDIG version 1.31 (Rohlf, 1999). Specimens damaged at or near any landmarks were discarded. Fluctuating asymmetry was measured in two ways on all specimens using landmarkbased multivariate geometric morphometrics. A multivariate size-based FA (FA 1 in Palmer and Strobek 2003) was calculated as ab solute value of (R L) or just R-L in signed FA distributions where R and L are th e centroid sizes of each wing (i.e., the sum of the distances of each landmark to their comb ined center of mass or centroid location). In addition, a multivariate shape-based measure of FA known as the Procrustes distance was calculated as the square root of the su m of all squared Euclidean distances between each left and right landmark after two-dime nsional Procrustes fitting of the data (Bookstein 1991; Klingenberg and McIntyre 19 98; FA 18 in Palmer and Strobeck 2003; Smith et al. 1997). Centroid size calcul ation, Euclidean distance calculation and Procrustes fitting were performed using yvind Hammers Paleontological Statistics program PAST version 0.98 (Hammer 2002). A sub-sample of 50 individuals from the fourth inbred line (pp4B3) was digitized five times to estimate meas urement error. In these cases, measures of FA were taken as the average FA value of the five replicate measurements for each specimen. Percent measurement error was also computed as

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49 (ME/average FA) x 100 where) 5 4 3 2 1 ( FA FA FA FA FA sd ME (as per Palmer and Strobek 2003). For assessing whether measur ement error (ME) interfered significantly with FA, a Procrustes ANOVA (in Micr osoft Excel) was performed on the five replications of the 50 specimen sub-samp le (Klingenberg and McIntyre 1998). Any subsequent statistical analys es were performed using SPSS Ba se 8.0 statistical software (SPSS Inc.). Model selection and inference The fits of unsigned size FA to three distributional models (half-normal, lognormal, and double Pareto lognormal (DPL N)) were compared in the Drosophila lines, by calculating negative log like lihoods and Akaike Informa tion Criteria (AIC) (Burnham and Anderson 1998; Hilborn and Mangel 1998) This method penalizes more complex models (those with more parameters) when selecting the best-fit di stributional model for a given dataset. Best fitting parameters were obtained by maximizing the log-likelihood function for each model (Appendix B). Th e maximization was performed using the conjugate gradient method within unconstrai ned solve blocks in the program MathCAD by MathSoft Engineering and Education Inc (2001). Results Model Simulation The amount of variance in termination times related directly to levels of simulated FA (i.e., low variance in termination time (t) gives low FA and vice versa). I found that not only does amount of FA increase with in creased variance in (t), but so do both kurtosis and the scaling effect in the distri bution tails. In Figure 3-6, normal quantile plots of signed FA are shown fo r different standard deviations in the normal variation of the termination of growth of geometrically expanding cell populations. The degree of the

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50 S or sigmoidal shape in the plot indicates level of leptokurtos is. The leptokurtosis in the quantile plot is reduced greatly with a decrea se in the standard deviation of the normal variation in growth termination times. SIGNED DISTRIBUTION OF FLUCTUATING ASYMMETRY DETRENDED FIT TO NORMAL 933 0 8 0 kurtosis634 0 5 0 kurtosis149 1 2 1 kurtosis812 11 0 3 kurtosis266 27 0 7 kurtosis Figure 3-6. Simulated distri butions of cell population size and FA for different amounts of variation in the termination of gr owth (variance in normally distributed growth stop time). The fit of simulate d data to the normal distribution can determined by how closely the plotted points follow the horizontal line (a good fit is horizontal).

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51 Experimental Results Both mean unsigned size FA and shape FA decreased with inbreeding in all lines. I also observed that the kurtosis of signed size FA and the skewness of both unsigned size and shape FA follow an identical trend. The trend was strongest in kurtosis, which decreased rapidly with inbreeding, indicating th at, as predicted, changes in mean FA are influenced strongly by the shape and tail behavior of the distribution of FA (Table 3-1). Table 3-1. Distribution parameters and m odel fit for multivariate FA in two wild populations and four inbred lines of Drosophila simulans and one isogenic line of Drosophila melanogaster Model fits are AIC for unsigned centroid size FA (zero is best fit, lowest number is next best fit). Kurtosis (signed size FA) Mean (unsigned size FA) Mean (shape FA) Scaling exponent in upper tail AIC HNORM AIC LGN AIC DPLN Wild 2004 28.3 .12 5.73 .38 0.0223 .0003 1.49 90.6 324 0.000 Wild 2005 45.9 .15 4.68 .19 0.0237 .0003 2.87 66.2 261 0.000 Inbred line 1 (OR18D) 2.43 .11 3.27 .09 0.0183 .0002 5.09 0.000 104 15.5 Inbred line 2(OR18D3) 8.85 .15 4.24 .13 0.0212 .0002 3.10 11.50 223 0.000 Inbred line 3 (PP4B2) 2.90 .15 3.31 .09 0.0184 .0002 4.63 0.000 130 44.0 Inbred line 4 (PP4B3) 3.30 .15 3.72 .11 0.0213 .0003 5.78 0.000 228 79.7 Isogenic (mel75) 0.550 .11 3.90 .10 0.0201 .0002 9.99 0.000 84.9 101 While wild populations on the whole, de monstrated increased mean, skew and leptokurtosis of FA compared to inbred lines, significant differences between populations were found in all distributional parameters even between each of the inbred lines (ANOVA mean size FA for a ll lines; F = 27.49, p < 0.001; ANOVA mean size FA for inbred lines only; F = 14.38 p < 0.001; note shape FA also show same result). Overall,

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52 inbred lines demonstrated lowe r kurtosis, just slightly above that expected from a normal distribution (Table 1). They also had lower mean FA. No significant differences were found with respect to these results according to sexes of flies. In Figure 3-7, I show the distribution of FA and detrended normal qua ntile plots for the wild population, four inbred lines and one isogen ic line respectively. As in the simulated data, the degree of th e S shape in the plot indicated level of leptokurtosis. The S shape in quantile plot is reduced greatly with inbreeding and nearly disappears in the isogenic line. Model Selection and Inference The comparison of candidate distributional m odels of FA demonstrated normalization associated with inbreeding in three of the four inbred lines (Table 3-1). In the remaining inbred line, the half-normal candidate mode l was a close second to the double Pareto lognormal distribution. In the wild populati on samples, the distribution of unsigned sizebased FA was best described by the double Pareto lognormal distribution (DPLN), a lognormal distribution with scali ng in both tails. In the best fitting parameters of this distribution there was no observable trend in lognormal mean or variance across wild populations and inbred lines. The sc aling exponent of the lower tail ( in Reed and Jorgensen 2004) was close to one in all lines while the scaling exponent in the upper tail ( in Reed and Jorgensen 2004) increased with inbreeding (Table 3-1). Both samples of the wild population demonstrat ed the lowest scaling expone nt in the upper tail of the distribution. The low sca ling exponents here indicate di vergence in variance (2004 and 2005 where < 3) and in the mean (2004 only where < 2). The inbred lines all show higher scaling exponents that are consistent with converging mean and variance.

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53 21 0 1 3 8 6 6 -.6 -7.8 -1 5 0 -2 2 2 2 1.0 1 3 .8 6 6 .6 7 .8 -15 .0 -22 .2 9 4 2 1 .0 1 3 8 6 6 -.6 -7.8 -1 5 0 -2 2 2 4 2 1 0 13 8 6 6 .6 7 8 1 5 0 2 2. 2 25. 6 2 0. 6 15 6 10.6 5 6 6 -4.4 -9. 4 -14.4 1 9. 4 -24 4 4 21.0 13.8 6.6 .6 7 .8 -15.0 -22.2 23. 1 15. 6 8.1 6 -6.9 -14. 4 -21 .9 9 4 25 28 2004 kurtosis wild 85 45 2005 kurtosis wild 43 2 18 kurtosis D OR inbred 30 3 3 4 kurtosis B PP inbred 90 2 2 4 kurtosis B PP inbred 85 8 3 18 kurtosis D OR inbred 55 0 75 kurtosis mel isogenic Figure 3-7. Distribution of fluctuating asymmetry and detr ended fit to normal for two samples of wild population collected in Gainesville, FL in summers of 2004 and 2005 and four inbred lines of Drosophila simulans derived from eight generations of full-sib crossing of the wild population of 2004. Also included is one isogenic line of Drosophila melanogaster (mel75). All n = 1000. The fit of data to the normal distributi on can determined by how closely the plotted points follow the horizontal line (a good fit is horizontal).

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54 The isogenic Drosophila melanogaster line demonstrated the highest scaling exponent and the most normalized dist ribution of unsigned FA. Measurement error was 7.6% for shape-based FA and 13.0% for centroid sizebased FA. In a Procrustes ANOVA (Klingenbe rg and McIntyre 1998) the mean squares for the interaction term of the ANOVA (MSInteraction) was highly significant p<0.001 indicating that FA variation was significantly larger than variation due to measurement error (ME). Discussion Revealing the Genetic Component of FA While we should be cautious about inferring process from pattern, the very similar results of both the modeling and the inbreeding experiment in Drosophila seem to suggest the presence of a scal ing component in the distribu tion of fluctuating asymmetry that is caused by a random multiplicative grow th process as suggested previously by Graham et al. 2003. This parameter appears to change with the genetic redundancy of the population which is presumably increased by genetic drift and and reduction in heterozygosity during inbreeding. Specifica lly, the scaling exponent (s) of the upper tail ( ) of the unsigned FA distribution, or outer tails of the signed FA distribution, are increased with inbreeding, causing more rapid power-law decay in the shape of the tails. This effect also reduces kurtosi s and apparently normalizes th e distribution of FA in more inbred populations. This sugge sts that individual genetic di fferences in the capacity to control variance in the term ination of random proportional growth (i.e., geometric Brownian motion) may be responsible for determining the shape and kurtosis of the distribution of FA In other words, leptokurtosis (kur tosis >3) in signed FA distribution

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55 indicates genetic variability in the populat ion while normality (kurtosis = 3) indicates genetic redundancy. Because leptokurtosis is very often obse rved in the distribution of FA, genetic variability potentially underlie s a large proportion of the vari ability observed in the FA of a given population. Observed differences or changes in FA are therefore not only a response of development to environmental stre ss, but clearly also can reflect inherent differences in the genetic redundancy of populatio ns. The significantly different levels of mean FA among the inbred lines in this st udy, presumably caused by the random fixation of certain alleles, also suggest s that there is a st rong genetic component to the ability to buffer development against random noise. It is assumed that the differences observed in FA between wild trapped and i nbred populations in this study do not indicate an effect of inbreeding depression in the study for two reasons. First, the four inbred lines analyzed in this study were vigorous in culture so the fixation of random alleles was probably not deleterious. Second, and more important, i nbreeding reduced FA rather than increasing it as would have been expect ed under genetic stress. It is also important to no te that kurtosis is potentially a much stronger indicator of FA than the distribution mean. The lo w scaling exponents found in the non-normal distributions of FA in the wild populations of Drosophila simulans are capable of slowing and perhaps even stopping, the conve rgence on mean FA with increased sample size. Because kurtosis is a fourth order mo ment, estimating it accurately also requires larger sample sizes. However, if kurtosis can be demonstrated to re spond as strongly to environmental stress as it does here to inbreedi ng, its potential streng th as a signal of FA may allow new interpretation of past studies of FA without the collect ion of more data.

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56 This may help resolve some of the current deba te regarding FA as a universal indicator of environmental health and as a potential sexua l signal in good genes models of sexual selection. Limitations of the Model There are certain aspects of the model presented here that may be oversimplifications of the real developmental process. First of all, this model assumes developmental instability is generated left-r ight growing tissue fiel ds with no regulatory feedback or control other than when growth is stopped. It is very likely that left-right regulation is able to occur at higher levels of organization (e.g., across multiple developmental compartments) even though ther e is no evidence of regulated cell cycling rates beyond the distance of 6-8 cells on average within any given developmental compartment (Milan et al. 1995). Therefor e this model explains how fluctuating asymmetry is generated, not how it is regulat ed. Second, this m odel only considers cell proliferation as influencing size. It is known that both cell size and programmed cell death, or apoptosis, are also important in regulating body size (Raff 1992, Conlin and Raff 1999). Both of these may play a more pr ominent role in vertebrate development, than they do in insect wings, where apparently growth is terminated during its exponential phase. Nevertheless, this simple model seems to replicate very well, certain behavioral aspects of the di stribution of fluctuating asym metry in the insect wing. The Sources of Scaling The basic process generating power-law s caling effects illustrated here (Reed 2001) offers an alternative perspective to the of ten narrow explanations of power-laws caused by self-organized criticality in the interactions among syst em components (Gisiger 2000). The power law that results from self-organiz ed criticality is created from the multiplicity

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57 of interaction paths in the network. As th e distance between two in teracting objects is increased in a network or multidimensional la ttice, the number of potential interaction pathways increases exponentially, while the correlation between such paths decreases exponentially (Stanley 1995). These opposing exponential relationshi ps create the power law scaling observed in simulati ons of self-organized critical systems. While there is no such interaction in our model, there is a power-law generated by opposing exponential functions. The constant degree of change represented by the power law in both the statistical physics of critical systems and the mathematics of both Reed and Jorgensens process and the model given here is the resu lt of the combined battle between both the exponentially increasing and decreasing rates of change (Reed 2001). While the natural processes are quite different, the underlying mathematical be havior is very similar. Potential Application to Cancer Screening Just as changes in the shape of the di stribution of fluctuating asymmetry is normalized across a population of genetically redundant individuals genetic redundancy in a population of cells may also help mainta in normal cell size and appearance. The loss of genetic redundancy in a tissu e is a hallmark of cancer. The abnormal gene expression and consequent genetic instab ility that characterizes cancerous tissue often results in asymmetric morphology in cells, tissues a nd tissue borders. Baish and Jain (2000) review the many studies connect ing fractal (scale free) ge ometry to the morphology of cancer. Cancer cells also are typically pleo morphic or more variable in size and shape than normal cells and this pleomorphy is associ ated with intercellula r differences in the amount of genetic material (Ruddon 1995). Frig yesyi et al. (2003) have demonstrated a power-law distribution of chromosomal aberra tions in cancer. Currently the type of distributional shape of the pleomorphic variabil ity in cell size is not known, or at least not

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58 published. However, Mendes et al. (2001) demo nstrated that cluster size distributions of HN-5 (cancer) cell aggregates in culture fo llowed a power-law scaled distribution. Furthermore these authors also demonstrated that in MDCK (normal) cells and Hep-2 (cancer) cells, cluster size distributions tr ansitioned from short-tailed exponential distributions to long-tailed power-law di stributions over time. The transition is irreversible and is likely an adaptive re sponse to high density and long permanence in culture due to changes in either control of replication or perhaps cell signaling. Taken collectively, these studies may suggest that scaling at higher levels of biological organization observed in cancer is due to incr eased relative differences in length of cell cycling rates of highly pleomorphic cell populations that have relatively larger intercellular differences in amount of geneti c material. The stochastic growth model I have proposed as the basis for higher vari ability in population level developmental instability or FA may also pr ovide a possible explanation for higher variability in the cell sizes of cancerous tissue. If genetic redundancy in growing tissue has the same distributional effect as gene tic redundancy in populations of organisms and tends to normalize the observed statistical distribution, then one might predict that the genetic instability of cancerou s tissue would create a scaling effect that causes pleomorphy in cells and scaling in cell cluster aggregati ons. Statistical comparison of cell and cell cluster size distributions in normal and can cerous tissues may provide a useful and general screening technique for detecting when genetic redundancy is compromised by cancer in normal tissues. Conclusion Until now, the basis of fluctuating asymmetry has been addressed only with abstract models of hypothetical cell signaling, or elsewhere, at the level of selection

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59 working on the organism with potential mechanism remaining in the black box. However, fluctuating asymmetry must first a nd foremost be envisioned as a stochastic process occurring during tissue growth, or in other words, occurring in an exponentially expanding population of cells. As demonstr ated in the model presented here, this expansion process can be represented by stocha stic proportional (geometric) growth that is terminated or observed randomly over time. These results imply that the fluctuating asymmetry observed in populations is not only related to poten tial environmental stressors, but also to a large degree, the unde rlying genetic variabili ty in those molecular processes that control the termination of growth. Therefore, fluctuating asymmetry responses to stress may be hard to interpre t without controlling fo r genetic redundancy in the population. Both the simulation and experi mental results suggest that measures of distributional shape like kurto sis, scaling exponent and tail weight may actually be a strong signal of variability in the underlying process that causes developmental instability. Therefore the kurtosis paramete r of the fluctuating asymmetry distribution may provide more information about fluc tuating asymmetry response than does a populations average or mean fluctuating asymmetry. This may provide a novel method by which to resolve conflicts in previous under-sampled research without the collection of more data.

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60 CHAPTER 4 TEMPERATURE RESPONSE OF FLUCTUAT ING ASYMMETRY TO IN AN APHID CLONE: A PROPOSAL FOR DETE CTING SEXUAL SELECTION ON DEVELOPMENTAL INSTABILITY Introduction Developmental instability is a potentially maladaptive component of individual phenotypic variation with some unknown basi s in both gene and environment (Mller and Swaddle 1997, Fuller and Houle 2003). De velopmental instability is most often measured by the manifestation of fluctuating asymmetry (FA), the ri ght minus left side difference in size or shape in a single trait across the population (Palmer and Strobek 1986, 2003; Parsons 1992, Klingenberg and McIntyre 1998). Because FA is thought to indicate stress during development, the primar y interest in the study of FA has been its potential utility as an indi cator of good genes in mate choice (Mller 1990, Mller and Pomiankowski 1993) or its util ity as a general bioindicato r of environmental health (Parsons 1992). The Genetic Basis of FA For FA to become a sexually selected trait, it must be assumed that it has a significant genetic basis, and can ther efore evolve (Mller 1990, Mller and Pomiankowski 1993). However, researcher s who use FA as a bioindicator of environmental health often assume that FA is a phenotypic response that is mostly environmental in origin (Parsons 1992, Lens et al. 2002). Studies of the heritability of FA seem to indicate low heri tability for FA exists (Whitlock and Fowler 1997, Gangestad

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61 and Thornhill 1999). However, because FA is essentially a variance that is often measured with only two data points per indi vidual, FA may have a stronger but less easily detectable genetic ba sis (Whitlock 1996, Fuller and Houle 2003). Additive genetic variation in FA in most studi es has been found to be minimal, but several quantitative trait loci studies suggest signi ficant dominance and character specific epistatic influences on FA (Leamy 2003, Leamy and Klingenberg 2005). Babbitt (chapter 3) has demonstrated that a populations genetic variab ility affects the distributional shape of FA. So while studies investigating mean FA may be inconclusive, changes in the populations distributional shape seem to indicate potential genetic in fluence on FA. However, no studies have observed FA in a clonal orga nism for the express purpose of assessing developmental instability that is purely environmental (i.e., nongenetic) in its origin (i.e., developmental noise). The Environmental Basis of FA It has long been assumed that FA is the re sult of some level of genetically-based buffering of additive independent molecular noise during development. Because of the difference in scale between the size of mol ecules and growing cells it would be unlikely that molecular noise would comprise an im portant source of variation in functioning cells. However, Leamy and Klingenberg ( 2005) rationalize that mo lecular noise could only scale to the level of tissue when developmentally important molecules exist in very small quantity (e.g., DNA or protein) and th erefore FA may repr esent a stochastic component of gene expression. This is some what similar in spirit to the Emlen et al. (1993) explanation of FA th at invokes non-linear dynamics of signaling and supply that may also occur during growth. Here FA is thought to be the result of the scaling up of

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62 compounding temporal asymmetries in signali ng between cells during growth. In this model, hypothetical levels of signaling compounds (morphogens) and or growth precursors used in the construction of cells vary randomly over time. When growth suffers less interruption, thus when it occurs faster, there is also less complexity (or fractal dimension) in the dynami cs of signaling and supply. Graham et al. (1993) suggest that nonlinear dynamics of hormonal signaling across the whole body may also play a similarly important role in the manifestation of FA. However, while the levels of FA are certainly influenced by the regul ation of the growth process, both Graham et al. (2003) and Babbitt (chapter 3) also suggest that FA le vels reflect noise during cell cycling that is amplified by exponentially expanding populat ions of growing cells. Although the proximate basis of FA is not well understood, its ultimate evolutionary basis, while heavily debate d, is easier to unde rstand. Mller and Pomiankowski (1993) first suggested that str ong natural or sexual selection can remove regulatory steps controlling the symmetri c development of certain traits (e.g., morphology used in sexual display). They sugge st that with respect to these traits (and assuming that they are somehow costly to pr oduce), individuals may vary in their ability to buffer against environmental stress and de velopmental noise in relation to the size of their individual energetic reserves; which are in turn often indicative of individual genetic quality. Therefore high genetic quality is associated with low FA. Most existing proximate or growth mechan ical explanations of FA assume that rapid growth is less stressful in the sense that fewer interruptions of growth by various types of noise should result in lower FA (Emlen et al. 1993, Graham et al. 1993). However, ultimate evolutionary explanations of FA assume that rapid growth is

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63 potentially more stressful because it is ener getically costly and therefore rapid growth should increase the level of FA when energy supply is limiting (Mller and Pomiankowski 1993). This high FA is relative and so should be especially prominent in individuals of lower genetic qu ality who can least afford to pay this additional energy cost. This fundamental difference between the predictions of proximate (or mechanistic) level and ultimate evolutionary level effects of temperature on FA is shown in Figure 1. The theoretical difference in the correlation of temper ature and growth rate to FA in both the presence and absence of energetic limitation could be used to poten tially detect sexual selection on FA. However it first should be confirmed th at FA should decrease with more rapid growth in the absence of en ergetic limitation to growth and genetic differences between individuals in a population. This later objective is the primary goal of this study. Temperature and FA in and Aphid Clone At a very basic level, entropy or noise in physical and chemical systems has a direct relationship with the physical ener gy present in the system. This energy is measured by temperature. Because FA is specu lated to tap into biol ogical variation that is somewhat free of direct genetic control, it may therefore res pond to temperature in simple ways. First, increased temperature may increas e molecular entropy which may in turn increase developmental noise during deve lopment thereby increasing FA. Second, increased temperature may act as a cue to shorten development time (as in many aphids where higher temperature reduces both body size and development time), thereby reducing the total time in which developmental errors may occur. This should reduce

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64 FA. A third possibility is that a species spec ific optimal temperature exists. If so, FA should increase while approaching both the upper and lower thermal tolerance limits of organisms. PROXIMATE LEVEL EFFECT ULTIMATE LEVEL EFFECTTEMPERATURE DEVELOPMENTAL TIME GROWTH RATE FLUCTUATING ASYMMETRY +-+-NON-OPTIMAL TEMPERATURE DEVELOPMENTAL TIME GROWTH RATE FLUCTUATING ASYMMETRY --+-ENERGETIC RESERVES STRESS + + +-Figure 4-1. Predicted proximate and ultimat e level correlations of temperature and growth rate to fluctuating asymmetry are different. Ultimate level (evolutionary) effects assume energetic limitation of individuals in the system. Proximate level (growth mechanical) e ffects do not. Notice that temperature and fluctuating asymmetry are negatively correlated in the proximate model while in the ultimate model they are positively correlated. Only a few studies have di rectly investigated the relationship between FA and temperature. The results are conflicting. FA is either found to increase on both sides of an optimal temperature (Trotta et al. 2005, Zakarov and Shchepotkin 1995), to be highest at low temperature (Chapman and Goulson 2000), to simply increase with increasing temperature (Savage and Hogarth 1999, Mpho et al. 2002) or not to respond

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65 (Hogg et al. 2001). None of these studies investigate the relationship between developmental noise (FA in a clonal line) and temperature. The characterization of developmental noise in response to temperature was investigated in this study using the cotton aphid, Aphis gossipyii. These aphids reproduce parthenogenetically, are not energetically lim ited in their diet (because they excrete excess water and sugar as honeydew) and pro duce wings that are easily measured using multiple landmarks. They demonstrate larg e visible variation in body size, wing size and even wing FA. The visible levels of wing as ymmetry in cotton aphids reflect levels of FA that are about four times higher than that ob served in other insect wings (Babbitt et al. 2006, Babbitt in press). Because parthenoge netic aphids cannot purge deleterious mutations each generation and because Florida clones often never use sexual reproduction to produce over-wintering eggs, this remarkably high FA may be the result of Mullers ratchet. Cotton aphids are al so phenotypically plastic in response to temperature, producing smaller lighter morphs at high temperatures and larger dark morphs at low temperatures. This feat ure allows observation of two genetically homogeneous groups in which different gene expression patterns (causing the color morphs) exist. The central prediction is that of the proximate effect model: that in the absence of energetic limitation and genetic variation, temperature and environmentally induced FA (or developmental noise) should be negatively correlated in a more or less monotonic relationship. Methods In March 2003, a monoclonal population of Cotton Aphids ( Aphis gossipyii Glover) was obtained from Dr. J. P. Michaud in Lake Alfred Florida and was brought to

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66 the Department of Entomology and Nematology at the University of Florida. The culture was maintained on cotton seedlings ( Gossipium ) grown at different temperatures (12.5C, 15C, 17C, 19C, 22.5C and 25C with n = 677 total or about 100+ per treatment) under artificial grow lights ( 14L:10D cycle). Because of potential under-sampling caused by a non-normal distribution of FA (see Ba bbitt et al. 2006), a second monoclonal population collected from Gainesville, FL in June 2004 was reared similarly but in much larger numbers at 12.5C, 15C, 17C, 19C, and 25C (n = 1677 or about 300+ per treatment). Development time for individual apterous co tton aphids (Lake Alfred clone) were determined on excised cotton leaf discs usi ng the method Kersting et al. (1999). Twenty randomly selected females were placed upon twenty leaf discs (5 cm diameter) per temperature treatment. Discs were set upon we t cotton wool in petri dishes and any first instar nymphs (usually 3-5) appearing in 24 hours were then left on the discs. Development time was taken as the average num ber of days taken to reach adult stage and compared across temperatures. Presence of shed exoskeleton was used to determine instar stages. Cotton was wetted daily and leaf discs were changed every 5 days. Humidity was maintained at 50%. In each temperature treatment, single clona l populations were allowed to increase on plants until crowded in order to stimulat e alate (winged individuals) production. Temperature treatments above 17C produced li ght colored morphs that were smaller and tended to feed on the undersides of leaves of cotton seedlings. Temperature treatments below 17C produced larger dark morphs that tended to feed on the stems of cotton seedlings. Alatae were collected using small brushes dipped in alcoho l and stored in 80%

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67 ethanol. Wings were dissected using fine in sect mounting pins and dry mounted as pairs on microscope slides. Species identificati on was by Dr. Susan Halb ert at the State of Florida Department of Plan t Industry in Gainesville. Specimens were dried in 85% ethanol, and then pairs of wings were dissected (in ethanol) and air-dried to the gl ass slides while ethanol evaporated. Permount was used to attach cover slips. This technique prev ented wings from floating up during mounting, which might slightly distort the landmark configuration. Dry mounts were digitally photographed. Six landmarks were identified as the two wing vein intersections and four termination points for the third subcostal. See Appendix A for landmark locations. Wing vein intersections were di gitized using TPSDIG version 1.31 (Rohlf, 1999). Specimens damaged at or near any landmarks were discarded. Fluctuating asymmetry was calculated using a multivaria te geometric morphometric landmark-based method. All landmarks are shown in Appendi x A. FA (FA 1 in Palmer and Strobek 2003) was calculated as absolute value of (R L) where R and L are the centroid sizes of each wing (i.e., the sum of the distances of each landmark to their combined center of mass or centroid location). In addition, a mu ltivariate shape-based measure of FA known as the Procrustes distance was calculated as the square root of the sum of all squared Euclidean distances between each left a nd right landmark after two-dimensional Procrustes fitting of the data (Bookstein 1991; Klingenberg and McIntyre 1998; FA 18 in Palmer and Strobeck 2003; Smith et al. 1997) This removed any difference due to size alone. Centroid size calculation, Euclidea n distance calculation and Procrustes fitting were performed using yvind Hammers Paleontological Statistics program PAST version 0.98 (Hammer 2002). Percent measur ement error was computed as (ME/average

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68 FA) x 100 where 3 / ) 3 1 3 2 2 1 ( FA FA FA FA FA FA ME in a smaller subset (200 wings each measured 3 times = FA 1, FA 2 and FA 3) of the total sample. All subsequent statistical analys es were performed using SPSS Ba se 8.0 statistical software (SPSS Inc.). Unsigned multivariate size and sh ape FA as well as the kurtosis of signed FA were then compared at various temperatures using one-way ANOVA. Results Development time (Figure 4-2) was very similar to previously published data (Kersting et al. 1999, Xia et al. 1999) decreasing monotonically at a much steeper rate in dark morphs than in light morphs. The distri butional pattern of centroid size, size FA and shape FA appear similar exhibiting right log skew distributions (Figure 4-3). Similar distributional patterns are obs erved within temperatures ( not shown) as that observed across temperatures (Figure 4-3). 8 10 12 14 16 18 20 22 24 26 5 10 15 20 25 30 DEVELOPMENT TIME (days) TEMPERATURE (C) Light Morph Dark Morph Figure 4-2. Cotton aphid mean developmen t time SE in days in relation to temperature (n= 531).

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69 2 80 .0 260.0 2 40. 0 2 20 .0 200.0 1 80. 0 1 60 .0 140.0 1 20. 0 1 00. 0 8 0 .0 6 0.0 4 0. 0 2 0 .0500 400 300 200 100 0 Std. Dev = 36.40 Mean = 104.3 N = 3644.00 CENTROID SIZE CENTROID SIZE FLUCTUATING ASYMMETRY 37.6 34.4 31.3 28.1 24.9 21. 8 18. 6 15. 4 12.2 9.1 5.9 2.7 -.5300 200 100 0 Std. Dev = 5.96 Mean = 7.7 N = 1822.00 .163 .150 .138 .125 .113 .100 .087 .075 .063 .050 .038 .025 .013200 100 0 Std. Dev = .02 Mean = .056 N = 1219.00 SHAPE FLUCTUATING ASYMMETRYFREQUENCY Figure 4-3. Distribution of isogenic size, size based and sh ape based FA in monoclonal cotton aphids grown in controlled en vironment at different temperatures. Distributions within each temperature treatment are similar to overall distributions shown here.

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70 Coefficient of variation for FA was sli ghtly higher for dark morphs (12.5 C = 92.59%, 15 C = 93.02%, 17 C = 88.95%, 19 C = 79.02%, 22.5 C = 80.00% and 25 C = 85.35%). Mean isogenic FA (both size and shape) was highly si gnificantly different across temperatures (ANOVA F = 6.691, df between group = 4, df within group = 1673, p < 0.001) in the Gainesville FL clone (Figur e 4-4) but not in the Lake Alfred clone (ANOVA F = 1.992, df between group = 5, df within group = 672, p < 0.078). This is an indication of undersampling in the Lake Alfr ed clone. In the Gainesville clone, mean centroid size FA (Figure 4-4A) and devel opment time (Figure 4-2) follow a nearly identical pattern, decreasing rapidly at first then slowing with increased temperature. TEMPERATURE (C) mean isogenic fluctuating asymmetry 10 12 14 16 18 20 22 24 26 0.052 0.054 0.056 0.058 0.06 0.062 0.064 0.066 0.068 10 12 14 16 18 20 22 24 26 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 A B Light Morph Light Morph Dark Morph Dark Morph Figure 4-4. Mean isogenic FA for (A.) centr oid sizebased and (B .) Procustes shapebased) in monoclonal cotton aphids (c ollected in Gainesville FL) grown on isogenic cotton seedlings at different temperatures.

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71 Mean shape FA was also significantly different across temperature classes (ANOVA F = 4.863, df within group = 4, df between group = 1673, p = 0.001) but this difference is due solely to elevated FA in the 12.5C group (Figure 4-4B). Less than one percent of the variation in FA was due to variation in body size (r = -0.101 for shape FA; r = 0.088 for size FA). Kurtosis in the shape of the distribution of size based FA (Figure 4-5) was significantly higher in dark morphs th an in light morphs (t = -2.21 p = 0.027). Within each morph (light or dark), kurtosis in the distribution of FA appears to increase with temperature slightly (Figure 4-5). Measurement error for shape FA was estimated at2.6% (Lake Alfred clone) and 2.2% (Gainesville clone). For size FA these estimates were 6.1% (Lake Alfred) and 5.7% (Gainesville). TEMPERATURE (C) KURTOSIS 10 12 14 16 18 20 22 24 26 -1 0 1 2 3 4 5 Light Morph Dark Morph Figure 4-5. Kurtosis of size-based FA in monoclonal cotton aphids grown on isogenic cotton seedlings at different temperatures.

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72 Discussion It appears that the prediction of the proximate effect model holds in the case of this population of cotton aphid, which, in general, is not energetically limited and in this study, is not genetically vari able. Temperature and growth rate (which are positively correlated in insects) are negatively associat ed with purely environmentally derived FA (i.e., developmental noise). This confirms the predictions of several proximate models of the basis of FA. Furthermore, it appears that centroid size-based FA is a simple function of development time. Because individual ge netic differences in the capacity to buffer against developmental noise are, in a sense, controlled for in this study by the use of natural clones, the response of FA to temp erature in this stud y represents a purely environmental response of FA. The fact that aphids excr ete large amount of water and sugar in the form of honeydew, as well as th e lack of a strong co rrelation between FA and size also suggests that there is no real energeti c cost to being large in the aphids in this study. An important next step will be to comp are this result to the association of growth rate to FA in genetically di verse and energetically limited sexually selected traits where the predicted association between growth ra te and FA would be the opposite of this study. It also appears that because the enviro nmental component of size-based isogenic FA is largely a function of developmental time and temperat ure, dark morphs of cotton aphids, which have much longer developm ent time than light morphs, also have significantly higher FA. The temperature trend in mean FA within both light and dark morphs, where development times are similar, is not consistent although it appears that the kurtosis of FA increases slightly with temperatur e within each morph.

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73 It is very interesting th at there is a strong differen ce in kurtosis between the two temperature morphs in this study. Previous work suggests that kurtosis is related to genetic variability in a populat ion (Babbitt in press). The difference observed here in light of genetic homogeneity in the monoclonal aphid cultures, suggests that there may be a difference in developmental stability of li ght and dark aphid mo rphotypes that is due primarily to the differential expression of genes in each phenotype. It is surprising that temperature trends in mean developmental noise are slightly different regarding whether a size or shapebased approach was used. In populations where individuals are genetically diverse, bot h size and shape-based measures of FA are often correlated to some degr ee (as they are here too). However, size and shape are regulated somewhat differently in that cell pr oliferation is mediated both extrinsically via cyclin E acting at the G1/S checkpoint of the cell cycle, predominantly affecting size, and intrinsically at via cdc25/string at the G2/M checkpoint, pred ominantly affecting pattern or shape (Day and Lawrence 2000). Extrin sic mechanisms regulate size through the insulin pathway and its associated horm ones (Nijhout 2003) provi ding a link between size and the nutritional envi ronment. This may explain why size FA follows development time more closely than shape FA in this study. Size is usually less canalized than is shape and therefore more variable. While this makes the size-based measure of FA generally attractive, it has been found in previous work (Babbitt et al. 2006) that population averag es of size-based FA are often so variable that they are frequently under-sampled b ecause of the broad and often long tailed distribution of FA. Shape-base d FA does not suffer as much from this problem and so its estimation is much better although it may be le ss indicative of environmental influences

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74 and perhaps therefore better suited for genetic studies of FA. Because size FA has been adequately sampled in this study and because size is more heavily influenced by environmental factors like temperature, I find that size-based FA is the more interesting measure of developmental inst ability in this study. In general, it is clear that developmen tal noise is not constant in genetically identical individuals cultured under near sim ilar environments. Th e overall distribution pattern in both size and shape FA is log skew ed even within temperature classes. This suggests that sampling error due solely to random noise can play a very significant role in FA studies. Furthermore, it appears that th e response of FA to the environment is potentially quite strong. This suggests that FA may indeed be a re sponsive bioindicator, however its response will not be very generalizable in enviro nments with fluctuation in temperature. In conclusion, the environmental respons e of developmental noise to temperature in absence of genetic variabi lity and nutrient limitation suppo rts an important prediction of theoretical explanations of the proximate basis of FA. This prediction is that FA should decrease with growth rate (and temper ature) in ectothermic organisms. Because it is only in sexually selected tr aits that the opposite predicti on should hold, that FA should increase and not decrease with growth rate, th e results presented here may offer a useful method for discriminating between FA that is under sexual selection an d FA that is not.

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75 CHAPTER 5 CONCLUDING REMARKS AND RECOMMENDATIONS In this dissertation, I confirm some exis ting hypotheses concerning FA and provide a new explanatory framework th at explains alteration in th e distribution of fluctuating asymmetry (FA) and its subsequent effect upon mean FA. The basis of FA and the influences that shape its response to gene s and the environment in populations can be suitably modeled by stochastic proportional growth in expanding populations of cells on both sides of the body that are terminated with a small degree of genetically-based random error. I model stochastic growth with geometric Brownian motion, a random walk on a log scale. And I also model er ror in terminating growth with a normal distribution. The resulting di stribution of FA is a lognorma l distribution characterized by power-law scaled tails. Because under a pow er-law distribution, increased sample size increases the chance of sampling rare events under these long tail s, convergence to the mean is potentially slowed or even stopped depending on the scaling exponents of the power-law describing the tails. I demonstrated that the effect of reduced convergence to the mean is substantial and has probably cause d much of the previous work on FA to be under-sampled (Chapter 1). I have also dem onstrated that the s caling effect in the distribution of FA is directly related to genetic variation in the population. Therefore a low scaling exponent and also high kurtosis is associated with a la rge degree of genetic variation between individuals in their ability to precisely terminate the growth process (Chapter 2). I also demonstrate that when genetic differences do not exist (where FA is comprised of only developmental noise) and when development is not limited

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76 energetically, such as in a parthenogenetic aphid clone, FA depends upon developmental time (Chapter 3). How Many Samples Are Enough? The answer to this clearly depends upon wh ether researchers chose to study FA in size or FA in shape. The required sample size depends upon the di stributional type and parameters of the distribution of FA. As pr eviously discussed, FA distributions differ depending upon whether FA is based on size or shape differences between sides. As demonstrated in chapter 1, multivariate m easures of FA based upon shape, such as Procrustes Distances, tend to ha ve distributional parameters that allow convergence to the mean that is about five times more rapid th an either univariate or multivariate measures of FA based upon size (Euclidian distance or ce ntroid size). While this information might seem to favor a shape-based appro ach, there are some ot her very important considerations given below. In the end, sa mple size must be independently evaluated in each study depending on the magnitude of asymmetry that one wishes to detect between treatments or populations. As a general ru le, if shape FA is used, 100-200 samples may suffice, but if size FA is used then many hundreds or even a thousand samples may be required to detect a similar magnitude difference. What Measure of FA is best? The handicap of sampling requirements of size-based FA aside, it appears that measures of the distributional shape of FA like kurtosis, scaling exponent in the upper tail and skewness are strongly related to si milar but smaller changes in mean size FA (chapter 2). Because size FA is most co mmonly used in past studies of FA, these measures of distribution shape might allo w clearer conclusions to be drawn from literature reviews and meta-a nalyses as well as individual studies of FA where the

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77 collection of more data is not possible. My work linking shape of the FA distribution to the genetic variability of the population will re quire that future studies of FA which are focused on the environmental component of FA be controlled for genetic differences between populations. Where FA is sought as a potential bioindicator of environmental stress, the potentially differing genetic structur e of populations will need to be considered in order to make meaningful conclusions about levels of FA. Additi onally, because in the absence of genetic variation in monoclonal aphi ds, size FA is directly related to the total developmental times of individuals (chapter 3), it may a better choice than shape FA for studies of environmental influences on FA. In most organisms, body size is less canalized than body shape. Body size is highly polygenic and depends upon many environmenta lly linked character traits. However body shape or patterning is determined by a sequential progression of the activity of far fewer genes. Body size is also regulated at a different checkpoint during the cell cycle than is body pattern formation or shape (Day and Lawrence 2000). Patterning or shape is regulated at G2/M checkpoint while size is regulated at G1 /S checkpoint. The latter checkpoint is associated with the insulin pa thway, linking size to nutrition and hence to the environment. Because of this the st udy of environmental responses of FA may ultimately be best served by using multivariate size-based measures of FA (i.e., centroid size FA) while at the same time making sure to collect enough samples to accurately estimate the mean. In conclusion, multivariate size FA may be the better metric for large studies interested in the eff ects of environmental factors upon FA. Where data is harder to come by (e.g., vertebrate field studies), multivariate shape FA may perform better with the

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78 caveat that shape and pattern are less likely to vary in a population than does size. Because centroid size must be calculated in order to derive Procrustes distance, both measures can easily and should be examined together. Does rapid growth stabilize or destabilize development? The effects of temperature and growth ra te on FA appear to support the proximate model which predicts that FA declines with increasing growth rates (Figure 4-3). Aphids from the Gainesville clone decrease mean FA in response to incr eased temperature and growth rate. It is likely that the Lake Alfred clone is under-sampled and therefore estimates of mean FA are not accurate in th at sample. The results of the Gainesville clone, which was sampled adequately, suggests that the prediction of proximate models of the basis of FA (Emlen et al. 1993, Gr aham et al. 1993), that rapid growth should decrease FA holds true. Mllers hypothesis that rapid growth is stressful because of incurred energetic costs does not appear to hold in the case of aphids. Of course aphids, being phloem feeders, generally have access to more water and carbon than they ever need. This is evidenced by analysis of honeydew composition in this and many other aphid species. Growth and reproduction in aphids is probably more limited by nitrogen than by water or energy (carbon). So it would seem that this system may not be a very good one in which to assess Mllers hypothe sis directly. It should be further investigated whether FA of sexually selected traits in energetically limited, genetically diverse populations is positively associated with growth rate as is predicted by the ultimate or evolutionary model presented in Chapter 4. Can fluctuating asymmetry be a sexually selected trait? Given that a genetically a ltered population (chapter 3) demonstrates a consistent response in the shape of the FA distribution does suggest that a poten tially strong genetic

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79 basis for FA exists, but is not easily observabl e through levels of mean FA. This implies some heritability exists and possibly opens th e door for selection to act upon FA in the context of mate choice. However, in the futu re it must be demonstr ated, with appropriate sample sizes, that the level of FA in th e laboratory can be altered by selection for increased or decreased FA. The opposing pred ictions of the proximate or mechanistic and ultimate (evolution through sexual select ion) explanations of FA, regarding the relationship of temperature, growth rate a nd FA may offer a method for detecting sexual selection upon FA (chapter 4) If FA has evolved as an indicator of good genes, and hence has related energetic costs, the expect ation that FA should increase with growth rate and temperature (in ectotherms) is plausi ble. Otherwise, if FA is caused by random accumulation of error during development, then the expectation is reversed. FA should decrease with growth rate and temperature as I have demonstrated in a clonal population of aphids that are not energe tically limited (chapter 4). Is fluctuating asymmetry a valu able environmental bioindicator? Given that the genetic structure of a population (chapter 3) has potentially strong effects upon the shape and location (mean) of th e distribution of FA, the usefulness of FA as a bioindicator of environmental stress has to be questioned. Average levels of FA of populations would be difficult to interpret or compare without additional information regarding genetic heterogeneity However, provided that the study of FA and population genetics were undertaken simultaneously, this problem could be avoided. So FA could still be of use in this regard, however its expense and ease of use compared to other bioindicators would need to be reassessed.

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80 Scaling Effects in Statistical Distributions: The Bigger Picture This dissertation demonstrates that underlying distributions of some biological data can contain partial self-similarity or power-l aw scaling. The proper model for describing data such as this sits at a midpoint between those models used in cl assical statistics and those of statistical physics: Le vy statistics (Bardou et al. 2003 ). To my knowledge, this work represents the first application of such a model in biology. The sources of power-law scaling in the natural sciences are diverse. Sornette (2003) outlines 14 different ways that power-law s can be created, some of which are very simple. The hypothesis that all power-law scaling in nature is due to a single phenomena such as self-organized criti cality (SOC) (Bak 1996, Gisiger 2000) or highly optimized tolerance (HOT) (Newman 2000) is unlikely. Reed (2001) suggests that the model I have used here, stochastic proportional growth th at is observed randomly, may explain a great deal of power-law scaled size distributions formerly speculated to have a single cause like SOC or HOT. These phenomena include di stributions of city size (Zipfs law), personal income (Paretos Law), sand particle size, species per genus in flowering plants, frequencies of words in sequences of text, size s of areas burnt in forest fires, and species body sizes, just to name a few examples. One of the most common ways that power-laws can be obtained is by combining exponential functions. This is effect is also observed when positive exponential/ geometric/proportional growth is observed w ith a likelihood described by to a negative exponential function such as in Reed and Jo rgensens (2004) model for generating size distributions. Power laws generated by combi ng exponential functions are also present in the statistical mechanics of highly interactiv e systems (e.g., SOC) where the correlation in behavior between two nodes or objects in an interaction network or lattice decreases

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81 exponentially with the distance between them while the number of potential interaction paths increases exponentially with the distance between th em (Stanley 1995). In Laplacian fractals, or diffusion limited aggregations observed in chemical electrodeposition and bacterial colony growth (Viscek 2001) there may be a similar interplay between exponential growth (doubling) and the diffusion of nutrients supporting growth which are governed by the no rmal distribution, of the form2xe y. The exponential function hol ds a special place in th e natural sciences. Any frequency dependent rate of cha nge in nature, or in other wo rds, any rate of change of something that is dependent on the proportion of that something present at that time, is described by the exponential f unction. Therefore it finds application to many natural phenomena including behavior of populations, chemical reactions, ra dioactive decay, and diffusion just to name a few. It seems only fitting that many of the power-laws we observe in the natural sciences probably owe their existence to the interplay of exponential functions, one of the most comm on mathematical relationships observed in nature. As far as we can ascertain from the record ing of ancient civili zations, human beings have been using numbers for at least 5000 year s if not longer. And yet the concepts of probability are a relatively recent human inventi on. The idea first appears in 1545 in the writings of Girolamo Cardano and is late r adopted by the mathematicians, Galileo, Fermat, Pascal, Huygens, Bernoul li and de Moivre in disc ussions of gambling over the next several hundred years. The concep ts of odds and of random chance are not generalized until the early tw entieth century by Andreyevic h Markov and not formalized into mathematics until the work of Andrei Kolmogorov in 1946.

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82 The role of uncertainty in nature is yet to be resolved. Does uncertainty lie only with us or does it underlie the very fabric of the cosmos? Most 19th century scientists (excepting perhaps Darwin) believed that th e universe was governed by deterministic laws and that uncertainty is solely due to human error. The first app lication of statistical distributions by Carl Frederick Gauss and Pierre Simon Laplace were concerned only with the problem of accounting for measuremen t error in astronomical calculations. The more recent view, that uncertainty is some thing real, was largely the work of early twentieth century quantum theorists Werner Heisenberg and Erwin Schrodinger and the statistical theory embodied in the work of Sir Ronald Fi sher. The basic conceptual revolution in modern physics was that observa tions cannot be made at an atomic level without some disturbance, therefore while one might observe something exactly in time, one cannot predict how the act of observing wi ll affect the observed in the future with any degree of certainty (at least at very small scales). This is Heisenbergs Uncertainty or Indeterminancy Principle. Scientists observe natural laws only through emergent properties of many atoms observed at vastly larger scal es where individual behavior is averaged into a collective whole. Jakob Bernoullis Law of Large Nu mbers, Abraham de Moivres bell shaped curve, and Sir Ronald Fishers estimation of th e mean are also examples of how scientists rely upon emergent properties of large system s of randomly behaving things in order to make sense of what is thought to be fundame ntally uncertain world. However, this probabilistic view of the natu ral world has been challenged in recent decades by some mathematicians who are again championing a dete rministic view of nature. Observations by Edward Lorenz, Benoit Mandelbrot and ot hers, of simple and purely deterministic

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83 equations that behave in complex unpredic table ways, has led to a revival in the deterministic view among mathematicians; in this case, uncertainty is a product of human inability to perceive systems that are extr emely sensitive to initia l starting conditions. For this reason, this new view is often calle d deterministic chaos. And so the question as to whether uncertainty is real or imagined is yet to be resolved by modern science. One remarkable observation of this latest revolution in mathematics is the frequent occurrence of scale invariance or power-law scaling in system s that exhibit this sort of complex and unpredictable behavior. And so ju st as the normal distribution or bell curve is an emergent property governing the random behavior of independent objects, the power-law appears to be an emergent propert y of nature as well; one that seems not only to often to appear in the beha vior of interacting objects (i .e., critical systems) but in systems where growth in randomly observed as well. Statistical physics now recognizes two classes of stable laws, one that leads to the Gaussi an or normal distribution and another that is Levy or power-law distributed. In the former class, emergent behavior is governed by commonly occurring random events wh ile in the latter cl ass the behavior of the group is governed by a few rarely occu rring random events. As I have already reviewed earlier, we find that convergence to the mean under these two frameworks can be radically different. Yet bot h are present in the natural world, and as I have shown in my work, both of these classes of behavior can underlie naturally occurring distributions, causing partial scale inva riance in real data. It is my convicti on that the biological sciences must in the future adopt the statistical methods of working under both of these frameworks and not simply make assumpti ons of normality whenever and wherever random events are found to occur.

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84 APPENDIX A LANDMARK WING VEIN INTERSECTIONS CHOSEN FOR ANALYSISOF FLUCTUATING ASYMMETRY Figure A1. Six landmark locations digitized for Aphis gossipyii Figure A2. Six landmark locations digitized for Apis mellifera

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85 Figure A3. Six landmark locations digitized for Chrysosoma crinitus Figure A4. Eight landmark locations digitized for Drosophila simulans

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86 APPENDIX B USEFUL MATHEMATICAL FUNCTIONS The following is a list of the probability density functions for candidate models of the distribution of fluctuating asymmetry. L og likelihood forms of these functions were maximized to obtain best fitting parameters of each model for our data. Asymmetric Laplace Distribution (see Kotz et al. 2001) )) ( ( 1 1) )( ( ) (2 xe x f for x )) ( ( 1 11 2) )( ( ) ( xe x f for < x where = location, = scale and = skew index (Laplace when = 1) Half-Normal Distribution 2 ( 2 1) )( ( 1 2) )( ( ) ( xe x f where = minimum data value and = dispersion Lognormal Distribution (see Evans 2000 or Limpert et al. 2001) ) 2 / ) (log ( 2 / 12 2) 2 ( 1 ) ( xe x x f where = location and = shape or multiplicative standard deviation Double Pareto Lognormal Distribution (see Reed and Jorgensen 2004) )) (log( ) (1x g x fx given the normal-Lap lace distribution ) ( ) ( ) ( ) ( y y yR R y g where the Mill s ratio R(z) is

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87 ) ( ) ( 1 ) ( z z z R and where is the cumulative density function and is probability density function for standard normal dist ribution N(0,1), where and are parameters that control power-law scaling in the tails of th e lognormal distribution. The limiting forms of the double Pareto logno rmal are the left Pareto lognormal ( ), right Pareto lognormal distributions ( ), and lognormal distributions ( ) with Pareto tails on only the left side, only the right side, or on neither side, respectively. A description of Reed and Jorgensens ge nerative model of double Pareto lognormal size distribution. Reed and Jorgensons (2004) generative model begins with the Ito stochastic differential equation representing a geom etric Brownian motion given below. Xdw Xdt dX with initial state X(0) = X0 distributed lognormally, log X0 N( 2). After T time units the state X( T ) is also distributed lognormally with log X( T ) N( + (2/2) T 2 + 2 T ). The time T at which the process is observ ed, is distributed with density f T(t) = et where is a constant rate. The double Pareto lognormal distribution is generated when geometric Brownian motion is sampled repeat edly at time t with a negative exponential probability.

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96 BIOGRAPHICAL SKETCH Gregory Alan Babbitt received a Bachelor of Arts in zoology from Ohio Wesleyan University in 1989. While working with Dr. Edward H Burtt, he co-authored Occurrence and Demography of Mites in Ea stern Bluebird and Tree Swallow Nests, Chapter 6 in Bird-Parasite Interactions, edited by J.E. Loye and M.Zuk. He worked for nine years as a zookeeper and aviculturi st at the Columbus Zoological Gardens, designing a successful breeding pr ogram for tropical storks, and, during that time, served on the Ciconiiformes Taxon Advisory Group for the American Zoo and Aquarium Association. He received his Master of Sc ience in 2000 from the University of Florida Department of Wildlife Ecology and Conservati on. He worked with Dr. Peter Frederick in designing baseline captive reproductiv e studies of Scarlet Ibises on Disneys Discovery Island Park that would co mplement the research groups ongoing investigations of reproductive fa ilure in ibises in and near Everglades National Park. In 2001, Greg began investigations regard ing the underlying basis and proper characterization of developmental instabilit y, the subject of this dissertation work presented here. The second chapter is also published in the February 2006 American Naturalist 167(2) pp230-245. The third chapter ha s been accepted for publication in a future volume of Heredity. After receiving his PhD in z oology from the University of Florida, Greg will be starting post-doctoral work with Dr. Yuseob Kim at Arizona State Universitys Biodesign Institute in the Cent er for Evolutionary Functional Genomics.